,transcrib]'s ,note3 ,?is book includes ext5sive ma!matical expressions & equations1 :i* c n always be easily repres5t$ 9 pla9 text4 ,! read] is enc\rag$ to d[nload ! ,,html v]sion ( ! text1 :i* repres5ts ! ma? m cle>ly4 ,= ! pla9 text v]sion1 ! foll[+ conv5tions are us$3 ,mix$ fractions are repres5t$ by a da% ) no spaces1 :ile subtraction is repres5t$ by a da% ) spaces on ei!r side4 ,= example3 #a-a_/b is "o & "o half4 #a - #a_/b is "o m9us "o half4 ,! sideways-#h symbol = inf9ity is repres5t$ z inf9ity4 ,squ>e1 cube1 & o!r roots are %[n by rais+ a quantity to ! appropriate fractional p[]4 ,= example3 #d7a_/b7 is ! squ>e root ( #d4 x7#a_/n7 is ! n? root ( x4 ,extra p>5!ses h be5 add$ z ne$$ to cl>ify ! correct ord] ( op]ations4 ,a ,,review ,,( ,,algebra ,,by ,,romeyn ,,h5ry ,,riv5burg1 ,a4,m4 ,,head ,,( ,,! ,,dep>tm5t ,,( ,,ma!matics ,,! ,,p$die ,,in/itute1 ,,hight/[n1 ,n4,j4 ,illu/ration ,,am]ican ,,book ,,company ,,new ,,york ,,c9c9nati ,,*icago ,,copyright1 #aiad1 ,,by ,,romeyn ,h4 ,,riv5burg4 ,,copyright1 #aiad1 ,,9 ,,great ,,brita94 ,a ,,review ,,( ,,algebra4 ,e4 ,p4 #f ,,preface ,9 mo/ high s*ools ! c\rse 9 ,elem5t>y ,algebra is f9i%$ by ! end ( ! second ye>4 ,by ! s5ior ye>1 mo/ /ud5ts h =gott5 _m ( ! pr9ciples1 & a ?or\gh review is necess>y 9 ord] to prep>e college c&idates = ! entrance exam9ations & = effective "w 9 ! fre%man ye> 9 college4 ,recogniz+ ?is ne$1 _m s*ools are devot+ at lea/ two p]iods a week = "p ( ! s5ior ye> to a review ( algebra4 ,= su* a review ! regul> textbook is inadequate4 ,f an emb>rassm5t ( ri*es ! tea*] f9ds x labori\s to select ! prop] examples1 :ile ! /ud5t wa/es "t 9 se>*+ = scatt]$ assignm5ts4 ,! object ( ?is book is to cons]ve ! "t & ef=t ( bo? tea*] & /ud5t1 by provid+ a ?or\gh & effective review t c readily be complet$1 if ne$ be1 9 two p]iods a week = a half ye>4 ,ea* /ud5t is expect$ to use 8 regul> textbook 9 algebra = ref]5ce1 z he w\ld use a diction>y1--to recall a def9ition1 a rule1 or a process t he has =gott54 ,he %\ld be enc\rag$ to ?9k 8 way out :]ev] possible1 h[ev]1 & to ref] to ! textbook only :5 =c$ to d s z a la/ resort4 ,! def9itions giv5 9 ! ,g5]al ,outl9e %\ld be review$ z occasion arises = _! use4 ,! :ole ,outl9e c be pr(itably employ$ = rapid class reviews1 by cov]+ ! "p ( ! ,outl9e t indicates ! answ]1 ! me?od1 ! example1 or ! =mula1 z ! case may be4 ,! :ole s*eme ( ! book is ord9>ily to h a page ( problems repres5t a day's "w4 ,?is1 ( c\rse1 does n apply to ! ,outl9es or ! few pages ( !ory1 :i* c be cov]$ m rapidly4 ,by ?is plan1 mak+ only a "p ( ! omissions indicat$ 9 ! next p>agraph1 ! ess5tials ( ! algebra c be readily cov]$1 if ne$ be1 9 f ?irty to ?irty-two lessons1 ?us leav+ "t = te/s1 ev5 if only eighte5 weeks1 ( two p]iods ea*1 are allott$ to ! c\rse4 ,if a brief c\rse is desir$1 ! ,miscellane\s ,examples 7pp4 #ca to #ce1 #ej to #eb71 _m ( ! problems at ! end ( ! book1 & ! ,college ,entrance ,exam9ations may be omitt$ )\t m>r+ ! cont9uity or ! compreh5siv5ess ( ! review4 ,,romeyn ,h4 ,,riv5burg4 ,,cont5ts ,,pages ,,outl9e ,,( ,,elem5t>y ,,& ,,int]m$iate ,,algebra #g-ac ,,ord] ,,( ,,op]ations1 ,,evaluation1 ,,p>5!ses #ad ,,special ,,rules ,,( ,,multiplication ,,& ,,division #ae ,,cases ,,9 ,,factor+ #af1 #ag ,,factor+ #ah ,,highe/ ,,common ,,factor ,,& ,,l[e/ ,,common ,,multiple #ai ,,fractions #bj ,,complex ,,fractions ,,& ,,fractional ,,equations #ba1 #bb ,,simultane\s ,,equations ,,& ,,involution #bc1 #bd ,,squ>e ,,root #be ,,!ory ,,( ,,expon5ts #bf-bh ,,radicals #bi1 #cj ,,miscellane\s ,,examples1 ,,algebra ,,to ,,quadratics #ca-ce ,,quadratic ,,equations #cf1 #cg ,,! ,,!ory ,,( ,,quadratic ,,equations #ch-da ,,outl9e ,,( ,,simultane\s ,,quadratics #db1 #dc ,,simultane\s ,,quadratics #dd ,,ratio ,,& ,,proportion #de1 #df ,,ari?metical ,,progression #dg ,,geometrical ,,progression #dh ,,! ,,b9omial ,,!orem #di ,,miscellane\s ,,examples1 ,,quadratics ,,& ,,beyond #ej-eb ,,problems--,,l9e> ,,equations1 ,,simultane\s ,,equations1 ,,quadratic ,,equations1 ,,simultane\s ,,quadratics #ec-eg ,,college ,,entrance ,,exam9ations #eh-hj ,,outl9e ,,( ,,elem5t>y ,,& ,,int]m$iate ,,algebra ,important ,def9itions ,factors2 coeffici5t2 expon5t2 p[]2 base2 t]m2 algebraic sum2 simil> t]ms2 degree2 homog5e\s expression2 l9e> equation2 root ( an equation2 root ( an expression2 id5tity2 conditional equation2 prime quantity2 highe/ common factor 7,h4 ,c4 ,f472 l[e/ common multiple 7,l4 ,c4 ,m472 involution2 evolution2 imag9>y numb]2 real numb]2 rational2 simil> radicals2 b9omial surd2 pure quadratic equation2 affect$ quadratic equation2 equation 9 ! quadratic =m2 simultane\s l9e> equations2 simultane\s quadratic equations2 discrim9ant2 symmetrical expression2 ratio2 proportion2 f\r? proportional2 ?ird proportional2 mean proportional2 ari?metic progression2 geometric progression2 ,s inf9ity ,special ,rules = ,multiplication & ,division #a4 ,squ>e ( ! sum ( two quantities4 7x y7#b4 #b4 ,squ>e ( ! diff]5ce ( two quantities4 7x - y7#b4 #c4 ,product ( ! sum & diff]5ce ( two quantities4 7s t77s - t74 #d4 ,product ( two b9omials hav+ a common t]m4 7x r77x m74 #e4 ,product ( two b9omials ^: correspond+ t]ms are simil>4 7#cx #bt77#bx - #et74 #f4 ,squ>e ( a polynomial4 7m - n_/#c k7#b4 #g4 ,sum ( two cubes4 7x#c y#c7_/7x y7 x#b - xy y#b4 #h4 ,diff]5ce ( two cubes4 7x#c - y#c7_/7x - y7 x#b xy y#b4 #i4 ,sum or diff]5ce ( two l p[]s4 7x#g y#g7_/7x y71 7x#e - y#e7_/7x - y71 7x#d - y#d7_/7x - y71 7x#d - y#d7_/7x y74 ,cases 9 ,factor+ #a4 ,common monomial factor4 mx my - mz m7x y - z74 #b4 ,tr9omial t is a p]fect squ>e4 x#b - #bxy y#b 7x - y7#b4 #c4 ,! diff]5ce ( two squ>es4 7a7 ,two t]ms4 x#b - y#b 7x y77x - y74 7b7 ,f\r t]ms4 x#b #bxy y#b - m#b 7x y m77x y - m74 7c7 ,six t]ms4 x#b #bxy y#b - p#b - #bpq - q#b 7x y7 7p q77x y7 - 7p q74 7d7 ,incomplete squ>e4 x#d x#b y#b y#d x#d #bx#b y#b y#d - x#b y#b 7x#b y#b xy77x#b y#b - xy74 #d4 ,tr9omial ( ! =m x#b bx c4 x#b - #ex #f 7x - #b77x - #c74 #e4 ,tr9omial ( ! =m ax#b bx c4 #bjx#b #gx - #f 7#dx #c77ex - #b74 #f4 ,sum or diff]5ce ( two cubes4 ,see ,special ,rules1 #g & #h4 two l p[]s4 ,see ,special ,rules1 #i4 #g4 ,common polynomial factor4 ,gr\p+4 t#b p t#b q - #bmp - #bmq t#b7p q7 - #bm7p q7 7p q77t#b - #bm74 #h4 ,factor ,!orem4 x#c #agx - #ah 7x - #a77x#b x #ah74 ,h4 ,c4 ,f4 & ,l4 ,c4 ,m4 a#b #b;a - #c 7a #c77;a - #a74 a#b #g;a #ab 7a #c77;a #d74 a#d #bg;a a7a #c77;a#b - #c;a #i74 ,h4 ,c4 ,f4 a #c4 ,l4 ,c4 ,m4 7a #c77;a - #a77;a #d7;a7a#b - #c;a #i74 ,fractions ,r$uction to l[e/ t]ms4 ,r$uction ( a mix$ numb] to an improp] fraction4 ,r$uction ( an improp] fraction to a mix$ numb]4 ,addition & subtraction ( fractions4 ,multiplication & division ( fractions4 ,law ( signs 9 division1 *ang+ signs ( factors1 etc4 ,complex fractions4 ,simultane\s ,equations ,solv$ by addition or subtraction4 sub/itution4 comp>ison4 ,graphical repres5tation4 ,involution ,law ( signs4 ,b9omial !orem laws4 ,expansion ( monomials & fractions4 b9omials4 tr9omials4 ,evolution ,law ( signs4 ,evolution ( monomials & fractions4 ,squ>e root ( algebraic expressions4 ,squ>e root ( ari?metical numb]s4 ,optional ,cube root ( algebraic expressions4 ,cube root ( ari?metical numb]s4 ,!ory ( ,expon5ts ,pro(s3 am x an a7m n72 7am7_/7an7 a7m - n72 7am7n a7mn72 a7mn77#a_/n7 am2 7a_/b7n 7an7_/7bn72 7abc7n an bn cn4 ,mean+ ( fractional expon5t4 z]o expon5t4 negative expon5t4 ,f\r rules ,to multiply quantities hav+ ! same base1 add expon5ts4 ,to divide quantities hav+ ! same base1 subtract expon5ts4 ,to raise to a p[]1 multiply expon5ts4 ,to extract a root1 divide ! expon5t ( ! p[] by ! index ( ! root4 ,radicals ,radical 9 its simple/ =m4 ,trans=mation ( radicals ,fraction "u ! radical sign4 ,r$uction to an entire surd4 ,*ang+ to surds ( diff]5t ord]4 ,r$uction to simple/ =m4 ,addition & subtraction ( radicals4 ,multiplication & division ( radicals a7#a_/n7 4 b7#a_/n7 ab7#a_/n74 7ab7#a_/n77_/7a7#a_/n77 b7#a_/n74 ,rationalization ,monomial d5om9ator4 ,b9omial d5om9ator4 ,tr9omial d5om9ator4 ,squ>e root ( a b9omial surd4 ,radical equations4 ,always *eck results to avoid extrane\s roots4 ,quadratic ,equations ,pure4 x#b a4 ,affect$4 ax#b bx c #j4 ,me?ods ( solv+ ,complet+ ! squ>e4 ,=mula4 ,develop$ f ax#b bx c #j4 ,factor+4 ,equations 9 ! quadratic =m4 ,prop]ties ( quadratics r#a -b_/#b;a 7b#b - #d;ac7#a_/b77_/7b;a74 r#b -b_/#b;a - 7b#b - #d;ac7#a_/b77_/7b;a74 ,!n r#a r#b -b_/a4 r#a 4 r#b c_/a4 ,discrim9ant1 b#b - #d;ac1 & its discussion4 ,nature or "* ( ! roots4 ,simultane\s ,quadratics ,,case ,i4 ,"o equation l9e>4 ,! o!r quadratic4 #cx - y #ab1 x#b - y#b #af4 ,,case ,,ii4 ,bo? equations homog5e\s & ( ! second degree4 x#b - xy y#b #ba1 y#b - #bxy -#ae4 ,,case ,,iii4 ,any two ( ! quantities x y1 x#b y#b1 xy1 x#c y#c1 x#c - y#c1 x - y1 x#b - xy y#b1 etc41 giv54 x#b y#b #da1 x y #i4 ,,case ,,iv4 ,bo? equations symmetrical or symmetrical except = sign4 ,usually "o equation ( high degree1 ! o!r ( ! fir/ degree4 x#e y#e #bdb1 x y #b4 ,,case ,v4 ,special ,devices ,i4 ,solve = a comp\nd unkn[n1 l xy1 x y1 7#a7_/7xy71 etc41 fir/4 x#by#b xy #f1 x #by -#e4 ,,ii4 ,divide ! equations1 memb] by memb]4 x#d - y#d #bj1 x#b - y#b #e4 ,,iii4 ,elim9ate ! quadratic t]ms4 #dx #cy #bxy1 #gx - #ey #exy4 ,ratio & ,proportion ,proportionals mean1 ?ird1 f\r?4 ,!orems #a4 ,product ( means equals product ( extremes4 #b4 ,if ! product ( two numb]s equals ! product ( two o!r numb]s1 ei!r pair1 etc4 #c4 ,alt]nation4 #d4 ,inv]sion4 #e4 ,composition4 #f4 ,division4 #g4 ,composition & division4 #h4 ,9 a s]ies ( equal ratios1 ! sum ( ! antec$5ts is to ! sum ( ! consequ5ts z any antec$5t1 etc4 ,special me?od ( prov+ f\r quantities 9 proportion4 ,let a_/b x1 a bx1 etc4 ,progressions ,developm5t ( =mulas4 l ar7n - #a74 l a 7n - #a7;d4 ,s 7arn - a7_/7r - #a74 ,s 7n_/#b77;a l74 ,s 7rl - a7_/7r - #a74 ,s 7n_/#b7b;a 7n - #a7;d4 ,sinf9ity 7a7_/7#a - r74 ,ins]tion ( means ,ari?metical4 ,geometrical4 ,b9omial ,!orem ,review ( b9omial !orem laws4 ,see ,involution4 ,expansion ( 7a b7n4 ,f9d+ any t]m by key numb] me?od4 r7?7 or 7r #a77?7 t]m me?od4 ,a ,,review ,,( ,,algebra ,,ord] ,,( ,,op]ations1 ,,evaluation1 ,,p>5!ses ,ord] ( op]ations3 ,fir/ ( all1 rais+ to a p[] & extract+ a root4 ,next1 multiplication & division4 ,la/ ( all1 addition & subtraction4 ,f9d ! value (3 #a4 #e 4 #bb - #be7a_/b7 _/ #e #bb 4 #h _/ #d - #b4 #b4 7#c x #f _/ #i7_/b - #bajj7a_/b7 _/ #e #d 4 #bc - 7#ad 4 #b7_/bh4 #c4 #i 4 #b _/ #f #c - #b 4 #db _/ #h7a_/c7 - #d 7#c 4 #bb7_/f4 ,evaluate3 #d4 7a#d - a#c b#c7_/7;a#b b#b7a_/b77 7ca7#a_/b7 a#c;bc7_/7abc71 if a #a1 b #b1 c #c4 #e4 t7#a_/c7 tm7#a_/c7 m7#a_/c71 if t #h1 m #bg4 #f4 7#bc #b;d a7#a_/b77_/7c;a b - cx - c7#a_/b77 77#c;c - d7x7_/7#g;ad - abc7#a_/b771 if a #e1 b #c1 c -#a1 d -#b1 x #j4 #g4 a - #e;b - a - 7#c;c - #c;b7 #b;c - #c7;a - #b;b - c71 if a -#c1 b #d1 c -#e4 7,yale47 ,simplify3 #h4 m - #bm - #cr - 7#dr - #bm74 #i4 #b;a - #e;d #c;c - 7a #b;d - #c;a #d;c74 #aj4 #c;c#b c7#b;a - #f;c - #c;a c - #d;a74 ,,special ,,rules ,,( ,,multiplication ,,& ,,division ,give results by inspection3 #a4 7g #a_/b k7#b4 #b4 7s - 7#bm7_/#c7b4 #c4 7#bv #cw77#bv - #cw74 #d4 7x #cts77x - #gts74 #e4 7#bl #c;g77#dl - #aa;g74 #f4 7a - 7#b;b7_/#c c - d7#b4 #g4 7x#c #hm#c7_/7x #bm74 #h4 7y#c - #bgk7#cm77_/7y - #ckm74 #i4 7c#e - d#e7_/7;c - d74 #aj4 7e#e d#e7_/7;e d74 #aa4 7x#d - y#d7_/7x - y74 #ab4 7x#d - y#d7_/7x y74 #ac4 7a - 4#jc77;a - 4#jjjg74 #ad4 7gn - #a_/b77;gn #c_/d74 #ae4 7t#g - v7#g_/b77_/7t - v7#a_/b774 #af4 7k#cb #a77k#af #a77k#h #a77k#d #a77k#b #a77k #a77k - #a74 #ag4 7a b7 7c d77a b7 - 7c d74 #ah4 7p - q r - s77p - q - r s74 #ai4 7#cm - n - l #br77#cm n - l - #br74 #bj4 7x #e77x - #b77x - #e77x #b74 #ba4 7a#b b#b - c - #b;d #c;e7#b4 #bb4 7s t - #bv_/#e #cw_/#f z#b7b4 #bc4 7x#e #cb7_/7x #b74 ,ref]5ces3 ,! *apt] on ,special ,rules ( ,multiplication & ,division 9 any algebra4 ,special ,rules ( ,multiplication & ,division 9 ! ,outl9e 9 ! front ( ! book4 ,,cases ,,9 ,,factor+ ,! numb] ( t]ms 9 an expression usually gives ! clue to ! possible cases "u :i* x may come4 ,by apply+ ! te/ = ea* & elim9at+ ! possible cases "o by "o1 ! "r case is readily f\nd4 ,h5ce1 ! numb] ( t]ms 9 ! expression & a ready & accurate k ( ! ,cases 9 ,factor+ are ! real keys to success 9 ?is vitally important "p ( algebra4 ,,case ,i4 ,a common monomial factor4 ,applies to any numb] ( t]ms4 #e;cx - #e;ct #e;cv - #ae c#b m #be c#c m#b #e;c7x - t v - #c;cm #e;c#b m#b74 ,,case ,,ii4 ,a tr9omial t is a p]fect squ>e4 ,?ree t]ms4 x#b - #bxm m#b 7x - m7#b4 ,,case ,,iii4 ,! diff]5ce ( two squ>es4 ,a4 ,two t]ms4 x#b - y#b 7x y77x - y74 ,b4 ,f\r t]ms4 x#b #bxy y#b - m#b 7x#b #bxy y#b7 - m#b 7x y m77x y - m7 ,c4 ,six t]ms4 x#b - #bxy y#b - m#b - #bmn - n#b 7x#b - #bxy y#b7 - 7m#b #bmn n#b7 7x - y7#b - 7m n7#b 7x - y7 7m n77x - y7 - 7m n74 ,d4 ,an incomplete squ>e4 ,?ree t]ms1 & #d? p[]s or multiples ( #d4 c#d c#b d#b d#d c#d #b;c#b d#b d#d - c#b d#b 7c#b d#b7b - c#b d#b 7c#b d#b cd77c#b d#b - cd74 ,,case ,,iv4 ,a tr9omial ( ! =m x#b bx c4 ,?ree t]ms4 x#b x - #cj 7x #f77x - #e74 ,,case ,v4 ,a tr9omial ( ! =m ax#b bx c4 ,?ree t]ms4 #bjx#b #gx - #f 7#dx #c77ex - #b74 ,,case ,,vi4 ,a4 ,! sum or diff]5ce ( two cubes4 ,two t]ms4 x#c y#c 7x y77x#b - xy y#b72 x#c - y#c 7x - y77x#b xy y#b74 ,b4 ,! sum or diff]5ce ( two l p[]s4 ,two t]ms4 x#d - y#d 7x - y77x#c x#by xy#b y#c72 x#e y#e 7x y77x#d - x#cy x#b y#b - xy#c y#d74 ,,case ,,vii4 ,a common polynomial factor4 ,any composite numb] ( t]ms4 t#b p t#b q - t#b r - g#b p - g#b q g#b r t#b 7p q - r7 - g#b 7p q -r7 7p q - r77t#b - g#b7 7p q - r77t g77t - g74 ,,case ,,viii4 ,! ,factor ,!orem4 ,any numb] ( t]ms4 x#c #agx - #ah 7x - #a77x#b x #ah74 ,,factor+ ,review ! ,cases 9 ,factor+ 7see ,outl9e on prec$+ pages7 & write out ! prime factors ( ! foll[+3 #a4 #h;a7#ac7 am7#ab74 #b4 x#g y#g4 #c4 #dx#b #aax - #c4 #d4 m#b n#b - 7#a #bmn74 #e4 -x#b #bx - #a x#d4 #f4 x7#af7 - y7#af74 7,five factors47 #g4 7x #a7b - #ex - #bi4 #h4 x#d x#b y#b y#d4 #i4 x#d - #aax#b #a4 #aj4 x7#bm7 #b #a_/7x7#bm774 #aa4 x7#fm7 #acx7#cm7 #ab4 #ab4 #d;a#b b#b - 7a#b b#b - c#b7b4 #ac4 7x#b - x - #f77x#b - x - #bj74 #ad4 a#d - #h;a - a#c #h4 #ae4 p#c #gp#b #adp #h4 #af4 #ah;a#b b #fj;ab#b #ej;b#c4 #ag4 x#c - #gx #f4 #ah4 #bd;c#b d#b - #dg;cd - #ge4 #ai4 7a#b - b#b7b - 7a#b - ab7#b4 #bj4 a#b x#c - 7#h;a#b7_/7y#c7 - x#c #h_/7y#c74 #ba4 gt - gk gl#b xt - xk xl#b4 #bb4 7m - n77#b;a#b - #b;ab7 7n - m77#b;ab - #b;b#b74 #bc4 a#b - x#b - y#b b#b #b;ab #bxy4 #bd4 7#b;c#b #c;d#b7;a 7#b;a#b #c;c#b7;d4 #be4 7n7n - #a77_/7a 4 #b7 a7n - #b7 b#b 7n7n - #a77n - #b77_/7a 4 #b 4 #c7 a7n - #c7 b#c4 #bf4 7x - x#b7c 7x#b - #a7c 7#a - x7#c4 7,m4 ,i4 ,t47 #bg4 7#bgy#c7b - #b7bgy#c77h;b#c7 7#h;b#c7b4 7,pr9ceton47 #bh4 7a#c #h;b#c77;a b7 - #f;ab7a#b - #b;ab #d;b#b74 7,m4 ,i4 ,t47 ,solve by factor+3 #bi4 x#c x4 #cj4 z#b - #dz - #de #j4 #ca4 x#c - x#b #dx - #d4 ,ref]5ce3 ,! *apt] on ,factor+ 9 any algebra4 ,,highe/ ,,common ,,factor ,,& ,,l[e/ ,,common ,,multiple ,def9e ,h4 ,c4 ,f4 & ,l4 ,c4 ,m4 ,f9d by factor+ ! ,h4 ,c4 ,f4 & ,l4 ,c4 ,m43 #a4 #cx#b - #cx1 #abx#b 7x#b - #a71 #ahx#c 7x#c - #a74 #b4 7x#b - #a77x#b #ex #f71 7x#b #cx77x#b - x - #f74 7,h>v>d47 #c4 x#b - y#b1 x#b y#b1 x#c y#c1 x#f y#f1 x#f - y#f4 7,college ,entrance ,bo>d47 #d4 x#c x#b - #b1 x#c #bx#b - #c4 7,cornell47 #e4 x#e - #bx#d x#b1 #bx#d - #dx#c - #dx #f4 7,yale47 #f4 x#b a#b - b#b #b;ax1 x#b - a#b b#b #b;bx1 x#b - a#b - b#b - #b;ab4 7,h>v>d47 #g4 #bx#b - x - #ae1 #cx#b - #aax #f1 #bx#c - x#b - #acx - #f4 7,college ,entrance ,bo>d47 #h4 7tv - v#b7c1 v#c - t#bv1 t#c - v#c1 v#b - #bvt t#b4 ,pick out ! ,h4 ,c4 ,f4 & ! ,l4 ,c4 ,m4 ( ! foll[+3 #i4 #h7x#b y77#cb7 7t#b z77#ai7 7m - n#c77ad71 #ab7x#b y77#bc7 7t#b z77#da7 7m - n#c77ag71 #ah7m - n#c77aa7 7x#b y77#ci7 7t#b z77#cg74 #aj4 #ag;ax#c 7y z77#aj7 7y - x77#ai7 7x z77#bg71 #cd;a#b x#d 7y z77#aa7 7y - x77#ba7 7x z77#ac71 #ea;a#c x#e 7y z7#d 7x z77#cb7 7y - x77#bi74 ,ref]5ce3 ,! *apt] on ,h4 ,c4 ,f4 & ,l4 ,c4 ,m4 9 any algebra4 ,,fractions ,def9e3 fraction1 t]ms ( a fraction1 reciprocal ( a numb]4 ,look up ! law ( signs z x applies to fractions4 ,except = ?is1 fractions 9 algebra are treat$ exactly ! same z !y are 9 ari?metic4 #a4 ,r$uce to l[e/ t]ms3 7a7 #cb_/bd2 7b7 7a#f - x#f7_/7;a#d - x#d72 7c7 7a b7#b - 7c d7#b_/7;a c7#b - 7b d7#b4 7,m4 ,i4 ,t47 #b4 ,r$uce to a mix$ expression3 7a7 #gef_/aa2 7b7 7a#c b#c7_/7;a - b74 #c4 ,r$uce to an improp] fraction3 7a7 #de-a_/h2 7b7 #i-aa_/ab qt42 7c7 a#b - ab b#b - 7b#c7_/7;a b74 ,add3 #d4 #e_/ah #g_/i #aa_/af #e_/h4 #e4 #e_/7a #bx7 - 7#cx7_/7#a - #bx7 7#d - #acx7_/7#dx#b - #a74 #f4 #a_/x7x - a77x - b7 #a_/;a7a - x77a - b7 #a_/;b7b - x77b - a74 ,multiply3 #g4 #gb_/aba x #ee_/ef x #gg_/ij4 #h4 7b - y7_/7a#c y#c7 x 7ca cy7_/7b#b by7 x 7b#f y#f7_/7;b#b y#b7 x b_/c4 ,divide3 #i4 7#ab_/be7 _/ 7#f_/ej74 #aj4 #a - 7ab7_/7a#b - ab b#b7 #a - 7ab7_/7a#b #b;ab b#b7 _/ 7a#c - b#c7_/7;a#c b#c74 7,yale47 #aa4 7x#d - y#d7_/7x#b - y#b7 _/ 7x y7_/7x#b - xy7 _/ 7x#b y#b7_/7x - y7 _/ 7x y7_/7xy - y#b74 7,%effield47 ,simplify3 #ab4 7#dy7_/x - 7#aey#b7_/7x#b7 #d _/ #d - 7#afy7_/x 7#aey#b7_/7x#b7 x #c - 7#dx #bjy7_/7#bx #ey74 ,ref]5ce3 ,! *apt] on ,fractions 9 any algebra4 ,,complex ,,fractions ,,& ,,fractional ,,equations ,def9e a complex fraction4 ,simplify3 #a4 7#c_/g #d_/e7_/7b - #c_/g 4 #d_/e74 #b4 7#b - #c_/b #b_/c7_/7e - #b_/c #c_/b74 #c4 #b - #b_/7a - #a_/a - #a_/7a #a_/b774 #d4 a_/7b#b7 - a_/b#b 7cb7_/7a - c_/b74 7,h>v>d47 #e4 ,if m #a_/7;a #a71 n #b_/7;a #b71 p #c_/7;a #c71 :at is ! value ( m_/7#a - m7 n_/7#a - n7 p_/7#a - p78 7,univ4 ( ,p5n47 #f4 ,simplify ! expression x y - #a_/x y - xy_/7x y77x#c - y#c7_/7x#b - y#b74 7,cornell47 #g4 ,simplify #a - 7#bxy7_/77x y7#b7_/a 7#bxy7_/77x - y7#b7 _/ 7#a - y_/x7_/7#a y_/x7#b4 #h4 ,solve 7#gy #i7_/d - y - 7#by - #a7_/i #g4 #i4 ,solve #b-a_/c - 7#b_/e77x#b #c7 7#ajx7_/#c #a - 7#bx#b7_/e4 #aj4 ,h[ mu* wat] mu/ be add$ to #hj p\nds ( a #e p] c5t salt solution to obta9 a #d p] c5t solution8 7,yale47 ,ref]5ce3 ,see ,complex ,fractions1 & ! fir/ "p ( ! *apt] on ,fractional ,equations 9 any algebra4 ,,fractional ,,equations #a4 ,solve = ea* lett] 9 turn #a_/;b #a_/p #a_/q4 #b4 ,solve & *eck3 7#ex #b7_/c - 7#c - 7#cx - #a7_/b7 7#cx #ai7_/b - 77x #a7_/f #c74 #c4 ,solve & *eck3 7#a_/b77x - a_/#c7 - 7#a_/c77x - a_/#d7 7#a_/d77x - a_/#e7 #j4 #d4 ,solve 7aft] look+ up ! special %ort me?od73 7#cx - #a7_/cj 7#dx - #g7_/ae x_/#d - 7#bx - #c7_/7abx - #aa7 7#gx - #ae7_/fj4 #e4 ,solve by ! special %ort me?od3 #a_/7x - #b7 - #a_/7x - #c7 #a_/7x - #d7 - #a_/7x - #e74 #f4 ,at :at "t betwe5 #h & #i o'clock are ! h&s ( a wat* 7a7 opposite ea* o!r8 7b7 at "r angles8 7c7 toge!r8 ,"w out 7a7 & /ate ! equations = 7b7 & 7c74 #g4 ,! =mula = conv]t+ a temp]ature ( ,f degrees ,fahr5heit into its equival5t temp]ature ( ,c degrees ,c5tigrade is ,c 7#e_/i77;,f - #cb74 ,express ,f 9 t]ms ( ,c1 & compute ,f = ! values ,c #cj & ,c #bh4 7,college ,entrance ,exam4 ,bo>d47 #h4 ,:at is ! price ( eggs :5 #b less = #bd c5ts raises ! price #b c5ts a doz58 7,yale47 #i4 ,solve #b_/7x - #b7 #b_/7d - x#b7 #e_/7x #b74 ,ref]5ce3 ,! ,*apt] on ,fractional ,equations 9 any algebra4 ,note p>ticul>ly ! special %ort me?ods1 usually giv5 ab\t ! middle ( ! *apt]4 ,,simultane\s ,,equations ,,note4 ,up to ?is po9t ea* topic pres5t$ has review$ to "s ext5t ! prec$+ topics4 ,= example1 factor+ reviews ! special rules ( multiplication & division2 ,h4 ,c4 ,f4 & ,l4 ,c4 ,m4 review factor+2 addition & subtraction ( fractions & fractional equations review ,h4 ,c4 ,f4 & ,l4 ,c4 ,m41 etc4 ,f ?is po9t on1 h[ev]1 ! int]dep5d5ce is n s m>k$1 & miscellane\s examples illu/rat+ ! "w already cov]$ w be giv5 v frequ5tly 9 ord] to keep ! :ole subject fre% 9 m9d4 #a4 ,solve by ?ree me?ods--addition & subtraction1 sub/itution1 & comp>ison3 #ex y #aa1 #cx #by #a4 ,solve & *eck3 #b4 #ab,r#a - #aa,r#b b #ab;c1 ,r#a ,r#b #b;b c4 #c4 7r - s7_/#b #be_/f - 7r s7_/#c1 7r s - #i7_/b - 7s - r - #f7_/c #j4 #d4 ,"o half ( ,a's m>bles exce$s "o half ( ,b's & ,c's toge!r by #b2 twice ,b's m>bles falls %ort ( ,a's & ,c's toge!r by #af2 if ,c _h f\r m m>bles1 he w\ld h "o f\r? z _m z ,a & ,b toge!r4 ,h[ _m has ea*8 7,college ,entrance ,bo>d47 #e4 ,! sides ( a triangle are a1 b1 c4 ,calculate ! radii ( ! ?ree circles hav+ ! v]tices z c5t]s1 ea* be+ tang5t ext]nally to ! o!r two4 7,h>v>d47 #f4 ,solve #bx #cy #g1 x - y #a graphically2 !n solve algebraically & comp>e results4 7,use coo]d9ate or squ>$ pap]47 ,factor3 #g4 x#d #d4 #h4 #b;d7#aj7 - #ajbd;d4 #i4 #b7x#c - #a7 - #g7x#b - #a74 ,ref]5ces3 ,! *apt]s on ,simultane\s ,equations & ,graphs 9 any algebra4 ,,simultane\s ,,equations ,,& ,,involution #a4 ,solve 7#c_/d7x - 7#e_/c7y #aa-a_/b1 7#e_/h7x - 7#c_/b7y #aj-a_/d4 ,look up ! me?od ( solv+ :5 ! unkn[ns are 9 ! d5om9ator4 ,%\ld y cle> ( fractions8 #b4 ,solve #a_/x - #a_/y - #a_/z #a_/;a1 #a_/y - #a_/z - #a_/x #a_/;b1 #a_/z - #a_/x - #a_/y #a_/;c4 #c4 ,solve graphically & algebraically #bx - y #d1 #bx #cy #ab4 #d4 ,solve graphically & algebraically #cx #gy #e1 #hx #cy -#ah4 ,review3 #e4 ,! squ>es ( ! numb]s f #a to #be4 #f4 ,! cubes ( ! numb]s f #a to #ab4 #g4 ,! f\r? p[]s ( ! numb]s f #a to #e4 #h4 ,! fif? p[]s ( ! numb]s f #a to #c4 #i4 ,! b9omial !orem laws4 7,see ,involution47 ,exp&3 7,indicate fir/1 !n r$uce47 #aj4 7b y7#g4 #aa4 7#b;a7_/#c - #ae4 #ab4 7x#b #b;a7#e4 #ac4 7x - y #bz7#c4 #ad4 ,a tra9 lo/ "o six? ( its pass5g]s at ! fir/ /op1 #be at ! second /op1 #bj ( ! rema9d] at ! ?ird /op1 ?ree qu>t]s ( ! rema9d] at ! f\r? /op2 #be rema94 ,:at 0 ! orig9al numb]8 7,m4 ,i4 ,t47 ,ref]5ces3 ,! *apt] on ,involution 9 any algebra4 ,also ! ref]5ces on ! prec$+ page4 ,,squ>e ,,root ,f9d ! squ>e root (3 #a4 #a #afm#f - #djm#d #ajm - #hm#c #bem#b4 #b4 7a#b7_/7x#b7 7#f;a7_/x #aa 7#fx7_/a 7x#b7_/7;a#b74 #c4 ,f9d ! squ>e root to ?ree t]ms ( x#b #e4 #d4 ,f9d ! squ>e root ( #ccg1efa4 #e4 ,f9d ! squ>e root ( #ahbc4bi4 #f4 ,f9d to f\r decimal places ! squ>e root ( #a4fgb4 7,pr9ceton47 #g4 ,add #b_/7x - #a7c #a_/7a - x7#b - #b_/7a - x7 - #a_/x4 #h4 ,f9d ! value (3 7#fd7a_/c7 4 #ab7_/bd _/ #b x #c - 7#b 4 #gb7_/7ad7 _/ #g x #a 7#a7a_/c7 4 #ag7_/7a 4 #ab7 - #d 4 #j4 #i4 ,simplify 7x y7#e 7x - y7#e7x y7#e - 7x - y7#e4 #aj4 ,solve by ! %ort me?od3 #e_/7g - x7 - 7#b-a_/d7x - #c_/d - 7x #aa7_/h 7#aax #e7_/af #j4 #aa4 ,x takes #c_/d ( a second = a ball to g f ! pit*] to ! cat*]1 & #a_/b ( a second = ! cat*] to h&le x & get (f a ?r[ to second base4 ,x is #ij feet f fir/ base to second1 & #acj feet f ! cat*]'s position to second4 ,a runn] /eal+ second has a />t ( #ac feet :5 ! ball leaves ! pit*]'s h&1 & beats ! ?r[ to ! base by #a_/h ( a second4 ,! next "t he tries x1 he gets a />t ( only #c-a_/b feet1 & is caught by #f feet4 ,:at is 8 rate ( runn+1 & ! velocity ( ! cat*]'s ?r[8 7,cornell47 ,ref]5ce3 ,! *apt] on ,squ>e ,root 9 any algebra4 ,,!ory ,,( ,,expon5ts ,review ! pro(s1 = positive integral expon5ts1 (3 ,i4 am x an a7m n74 ,,ii4 7am7_/7an7 a7m - n74 ,,iii4 7am7n a7mn74 ,,iv4 a7mn77#a_/n7 am4 ,v4 a_/bn 7an7_/7bn74 ,,vi4 7abc7n an bn cn4 ,to f9d ! mean+ ( a fractional expon5t4 ,assume t ,law ,i holds = all expon5ts4 ,if s1 a7#b_/c7 4 a7#b_/c7 4 a7#b_/c7 a7#f_/c7 a#b4 ,h5ce1 a7#b_/c7 is "o ( ! ?ree equal factors 7h5ce ! cube root7 ( a#b4 ,!re=e a7#b_/c7 a#b7a_/c74 ,9 ! same way1 a7#d_/e7 4 a7#d_/e7 4 a7#d_/e7 4 a7#d_/e7 4 a7#d_/e7 a7#bj_/e7 a#d4 ,h5ce1 a7#d_/e7 is "o ( ! five equal factors 7h5ce ! fif? root7 ( a#d4 ,!re=e a7#d_/e7 a#d7a_/e74 ,9 ! same way1 9 g5]al1 a7p_/q7 ap7#a_/q74 ,h5ce1 ! num]ator ( a fractional expon5t indicates ! p[]1 ! d5om9ator indicates ! root4 ,to f9d ! mean+ ( a z]o expon5t4 ,assume t ,law ,,ii holds = all expon5ts4 ,if s1 7am7_/7am7 a7m - m7 a#j4 ,b by division1 7am7_/7am7 #a4 ,!re=e a#j #a4 ,axiom ,i4 ,to f9d ! mean+ ( a negative expon5t4 ,assume t ,law ,i holds = all expon5ts4 ,if s1 am x a7-m7 a7m - m7 a#j #a4 ,h5ce1 am x a7-m7 #a4 ,!re=e a7-m7 #a_/7;am74 ,rules3 ,to multiply quantities hav+ ! same base1 add expon5ts4 ,to divide quantities hav+ ! same base1 subtract expon5ts4 ,to raise a quantity to a p[]1 multiply expon5ts4 ,to extract a root1 divide ! expon5t ( ! p[] by ! index ( ! root4 #a4 ,f9d ! value ( #cb - #e x #dj #h7-b_/c7 #a7b_/e74 #b4 ,f9d ! value ( #h7-b_/c7 #i7c_/b7 - #b7-b7 #a7-b_/e7 - #gj4 ,give ! value ( ea* ( ! foll[+3 #c4 7#cj7_/e1 #c_/7ej71 7#cj7_/7ej71 #cj x #e1 #c x #ej1 #cj x #ej1 #cj #ej1 #cj - #ej4 #d4 ,express #gj z "s p[] ( #g divid$ by itself4 ,simplify3 #e4 #af7a_/c7 4 #b7a_/b7 4 #cb7e_/f74 7,*ange to ! same base fir/47 #f4 #b_/7h7-c777a_/e74 #g4 7xn77n #b7_/7x7n #a777x7n - #a774 #h4 7x #cx7#b_/c7 - #bx7#a_/c777c - #bx7-#a_/c7 #dx7-#b_/c774 #i4 7a#b;b7_/7c#b;d77#a_/b7 x 7c#c;d7_/7ab#c77a_/c7 x 7a7#a_/c7;c7_/7b7#a_/d7;d7#e_/ab77b4 #aj4 7a7-#d77_/7;b7-#b7;c77-#c_/d7 x 7a7-#a7;bc7-#c77a_/b77_/7;ab7-#a777a_/b7 4 #aa4 7a#b7a_/c77_/7;b7-#a77a_/d77 4 7c7-#c77a_/b77_/7;a7#a_/c77 4 7b7-#a_/d7;a7#a_/c77_/7;c7-#a777-f74 ,ref]5ce3 ,! *apt] on ,!ory ( ,expon5ts 9 any algebra4 ,solve = x3 #a4 x7#b_/c7 #d4 #b4 x7-#c_/d7 #h4 ,factor3 #c4 x7#b_/c7 - #i4 #d4 x7#c_/e7 #bg4 #e4 x7#b;a7 - y7-#f74 #f4 a7#a_/c7 x7#a_/b7 - #c;a7#a_/c7 #ex7#a_/b7 - #ae4 #g4 ,f9d ! ,h4 ,c4 ,f4 & ,l4 ,c4 ,m4 ( a#b a7#c_/b7 b7#a_/b7 a7#a_/b7 b7#c_/b7 - b#b1 a#b - a7#c_/b7 b7#a_/b7 - a7#a_/b7 b7#c_/b7 - b#b4 #h4 ,simplify ! product (3 7ayx7-#a777a_/b71 7bxy7-#b777a_/c71 & 7y#b;a7-#b7;b7-#b777a_/d74 7,pr9ceton47 #i4 ,f9d ! squ>e root (3 #be;a7#d_/c7;b7-#c7 - #aj;a7#b_/c7;b7-#c_/b7 - #di #aj;a7-#b_/c7;b7#c_/b7 #be;a7-#d_/c7;b#c4 #aj4 ,simplify 7#b7n #b77_/7d7-n77 _/ 7#hn7_/7#bc77a_/e74 #aa4 ,f9d ! value ( 7#g 4 #acj _/ #g7_/7baj7 #cj x 7#dj 4 #gj7_/7g;a b7#j #h7-b_/c74 #ab4 ,express z a p[] ( #b3 #hc2 #de2 #dc 4 #h7b_/c7 4 #af7c_/d74 #ac4 ,simplify 7x7a #a77_/7x7#a - a77a _/ 7xa7_/7x7#a - a777a - #a77a_/7c;a - #a774 #ad4 ,simplify 7x7#e_/b7 y7#d_/c77_/7z7-#e_/d77 4 7z#d7_/7x7-#c7 y7-#e_/c77 _/ 7y7-#b7 z7#a_/d77_/7x7-#a_/b777a_/e74 #ae4 ,exp& 7a7#a_/b7 b7#a_/c77d1 writ+ ! result ) fractional expon5ts4 ,ref]5ce3 ,! *apt] on ,!ory ( ,expon5ts 9 any algebra4 ,,radicals #a4 ,review all def9itions 9 ,radicals1 also ! me?ods ( trans=m+ & simplify+ radicals4 ,:5 is a radical 9 its simple/ =m8 #b4 ,simplify 7to simple/ =m73 #b_/c7a_/b72 #a_/aa7a_/b72 #c_/e7a_/c72 #ce_/f7a_/b72 7#b;a_/b77#h;b#b7_/7bg;a77#a_/b72 #e_/7xn77#a_/bn72 7a b7#b 7-a#d7_/77;a b7#e77a_/c72 #bg7a_/b72 #ed7a_/c72 -#eabe7a_/b74 #c4 ,r$uce to entire surds3 #bc7a_/b72 #bc7a_/d72 #fb7a_/c72 ab#b7a_/n72 -#cb7a_/c72 #c;a7a #b7_/7f;a#b77a_/c72 7a #by77a - #by7_/7a #by77#a_/b74 #d4 ,r$uce to radicals ( l[] ord] 7or simplify indices73 a#b7a_/d72 a#c7a_/f72 #bg;a#c7a_/f72 #ha a#d x#h7a_/ab72 #ix#b y#d z#aj7a_/bn74 #e4 ,r$uce to radicals ( ! same degree 7ord]1 or index73 #g7a_/b7 & #aa7a_/c72 #e7a_/c7 & #c7a_/d72 #g7a_/f7 & #c7a_/b72 xm7#a_/n7 & xn7#a_/m72 cy7#a_/x71 cz7#a_/y71 & cx7#a_/z74 #f4 ,:i* is great]1 #c7a_/b7 or #d7a_/c78 #bc7a_/c7 or #bb7a_/b78 #g4 ,:i* is greate/1 #c7a_/b71 #e7a_/c71 or #g7a_/d78 ,give "w & arrange 9 desc5d+ ord] ( magnitude4 ,collect3 #h4 #abh7a_/b7 - #bej7a_/b7 #gb7a_/b7 - #ah7a_/b74 #i4 #be_/c7a_/b7 7#a_/f7fj7a_/b7 #ae7a_/b7 #c_/e7a_/b74 #aj4 7m - n7#b;a7#a_/b7 7m n7#b;a7#a_/b7 - am#b7a_/b7 a7n - m7#b7a_/b7 - a7#a_/b74 #aa4 ,a & ,b ea* %oot ?irty arr[s at a t>get4 ,b makes twice z _m hits z ,a1 & ,a makes ?ree times z _m misses z ,b4 ,f9d ! numb] ( hits & misses ( ea*4 7,univ4 ( ,cal47 ,ref]5ce3 ,! *apt] on ,radicals 9 any algebra 7fir/ "p ( ! *apt]74 ,! mo/ important pr9ciple 9 ,radicals is ! foll[+3 7ab77#a_/n7 a7#a_/n7 b7#a_/n74 ,h5ce ab7#a_/n7 a7#a_/n7 4 b7#a_/n74 ,or1 a7#a_/n7 4 b7#a_/n7 ab7#a_/n74 ,f ?is also 7ab7#a_/n77_/7a7#a_/n77 b7#a_/n74 ,multiply3 #a4 #bd7a_/c7 by #cf7a_/c74 #b4 #b7a_/b7 by #c7a_/c74 #c4 #b7a_/d7 by #d7a_/f74 #d4 a x7#a_/b77a_/b7 by a - x7#a_/b77a_/b74 #e4 #b7a_/b7 #c7a_/b7 - #e7a_/b7 by #b7a_/b7 - #c7a_/b7 #e7a_/b74 #f4 -p_/#b 7p#b - #dq7#a_/b77_/b by -p_/#b - 7p#b - #dq7#a_/b77_/b4 ,divide3 #g4 #bg7a_/b7 by #c7a_/b74 #h4 #dah7a_/b7 by #ecb7a_/b74 #i4 #cab7a_/c7 by #f7a_/b74 #aj4 #c7a_/b7 by #c7a_/d74 #aa4 #faje7a_/b7 #ahdj7a_/b7 - #deab7a_/b7 by #cae7a_/b74 7,%ort division47 #ab4 #ajah7a_/c7 - #dfj7a_/c7 #eajj7a_/c7 by #ccj7a_/c74 ,rationalize ! d5om9ator3 #ac4 #b_/7c7a_/b772 #g_/7g7a_/b772 #e_/7be7a_/b772 #c_/7;a#b7a_/e772 #d_/7;a#c7a_/g774 #ad4 #b_/7b7a_/b77 #c7a_/b772 7a7#a_/b7 b7#a_/b77_/7;a7#a_/b7 - b7#a_/b772 #c_/7c - #c7a_/b774 #ae4 #c7a_/b7 #b7a_/b7_/f7a_/b7 #c7a_/b7 - #b7a_/b74 ,review ! me?od ( f9d+ ! squ>e root ( a b9omial surd4 7,by inspection pref]ably47 ,!n f9d squ>e root (3 #af4 #e #bf7a_/b74 #ag4 #ag - #abb7a_/b74 #ah4 #g - #cc7a_/b74 ,ref]5ce3 ,! *apt] on ,radicals 9 any algebra1 beg9n+ at ,addition & ,subtraction ( ,radicals4 ,,miscellane\s ,,examples1 ,,algebra ,,to ,,quadratics ,results by inspection1 examples #a-aj4 ,divide3 #a4 7x7#e_/ag7 y7#e_/ag77_/7x7#a_/ag7 y7#a_/ag774 #b4 7x - y7_/7x7#a_/c7 - y7#a_/c774 #c4 7m#b n#b7_/7m7#b_/c7 n7#b_/c774 #d4 7x - y#b7_/7x7#a_/c7 - y#b7a_/c774 ,multiply3 #e4 a7-#c_/d7 #b_/7m7#a_/b77b4 #f4 7,k7-#b_/g7 - g7-#aa_/be77b4 #g4 7r7#bs7 l7-#cm777r7#bs7 - l7-#cm774 #h4 a7-#b7 b7-#c7 - #a_/7;c#b7b4 #i4 7#c,kx #dt7-#c777c,kx - #gt7-#c774 #aj4 7#by7#b_/g7 - #dj,k#c77cy7#b_/g7 #ee,k#c74 ,factor3 #aa4 x7#b_/c7 - #fd4 #ab4 y7#c_/e7 #bg4 #ac4 b7#c_/b7 - #hm7-#a74 #ad4 #cp - #hp7#a_/b7 - #ce4 ,factor1 us+ radicals in/ead ( expon5ts3 #ae4 #fj - #gc;b7#a_/b7 - #f;b4 #af4 #aem - #bmn7#a_/b7 - #bdn4 #ag4 a - b 7factor z diff]5ce ( two squ>es74 #ah4 a - b 7factor z diff]5ce ( two cubes74 #ai4 a - b 7factor z diff]5ce ( two f\r? p[]s74 #bj4 ,f9d ! ,h4 ,c4 ,f4 & ,l4 ,c4 ,m4 ( x#b xy7#a_/b7 - #by1 #bx#b #exy7#a_/b7 #by1 #bx#b - xy7#a_/b7 - y4 #ba4 ,solve 7%ort me?od7 7x - #g7_/7x - #h7 - 7x - #h7_/7x - #i7 7x - #d7_/7x - #e7 - 7x - #e7_/7x - #f74 #bb4 ,simplify 7ab_/c bc_/a ca_/b7_/7a_/bc b_/ca c_/ab7 x 77a b c7#b7_/7;ab bc ca7 - #b4 7,pr9ceton47 #a4 ,solve = p3 #b7p - #c7 #abh4 #b4 ,solve = t3 t7#c_/b7 -#bg4 #c4 ,f9d ! squ>e root ( #haad4djfd4 ,:at1 !n1 is ! squ>e root ( 4#jjhaaddjfd8 ( #haaddj4fd8 ,f any ( ! above c y det]m9e ! squ>e root ( 4#jhaaddjfd8 #d4 ,! ,h4 ,c4 ,f4 ( two expressions is a7a - b71 & _! ,l4 ,c4 ,m4 is a#b;b7a b77a - b74 ,if "o expression is ab7a#b - b#b71 :at is ! o!r8 #e4 ,solve 7%ort me?od73 #e_/7g - x7 - 7#b-a_/d7x - #c_/d - 7x #aa7_/h 7#aax #e7_/af #j4 #f4 ,solve #b_/m - #c_/n #aj_/p -#c1 #d_/m #e_/p #f_/n #ae1 #a_/m - #a_/n #e_/p -#a_/b4 #g4 ,simplify #bab_/c7a_/b7 - #ed_/e7a_/b7 #fd-a_/f7a_/b7 - #ajc-a_/e7a_/b7 7#dj_/c7aa-a_/d7a_/b74 #h4 ,does #af x #be7a_/b7 #d x #e8 ,does #af #be7a_/b7 #d #e8 #i4 ,write ! fraction #e_/7d #bc7a_/b77 ) rational d5om9ator1 & f9d its value correct to two decimal places4 #aj4 ,simplify 7p p#b - q7#a_/b7_/b77a_/b7 7p - p#b - q7#a_/b7_/b77a_/b7b_/p q7#a_/b74 7,pr9ceton47 #a4 ,rationalize ! d5om9ator ( #f7a_/b7 #c7a_/b7 - #cb7a_/b7_/f7a_/b7 - #c7a_/b7 #cb7a_/b74 7,univ4 ( ,cal47 #b4 ,simplify #b7n #d7 - #b7bn7_/#b7b7n #c774 7,univ4 ( ,p5n47 #c4 ,f9d ! value ( #a #h7-x_/#c7_/7hx77#a_/b7 #aj7x - #b71 :5 x #b4 7,cornell47 #d4 ,f9d ! value ( x if x7#f_/e7 y#d1 y7#b_/c7 #i4 7,m4 ,i4 ,t47 #e4 ,a fi%]man told a y>n ab\t a fi% he _h caught4 ,if ! fi% 7 half z long z he said x 01 x w\ld be #aj in*es m ?an twice z long z x is4 ,if x 7 #d in*es long] ?an x is1 & he _h fur!r exagg]at$ its l5g? by add+ #d in*es1 x w\ld be #a_/e z long z he n[ said x 04 ,h[ long is ! fi%1 & h[ long did he fir/ say x 08 7,m4 ,i4 ,t47 #f4 ,! =ce ,p necess>y to lift a weight ,w by means ( a c]ta9 ma*9e is giv5 by ! =mula ,p a b,w1 :]e a & b are con/ants dep5d+ on ! am\nt ( friction 9 ! ma*9e4 ,if a =ce ( #g p\nds w raise a weight ( #bj p\nds1 & a =ce ( #ac p\nds w raise a weight ( #ej p\nds1 :at =ce is necess>y to raise a weight ( #dj p\nds8 7,fir/ det]m9e ! con/ants a & b47 7,h>v>d47 #g4 ,r$uce to ! simple/ =m3 #d_/b7n #b77a_/n72 ax7a7-#a7x - ax7-#a77_/x7#b_/c7 - a7#b_/c74 #h4 ,det]m9e ! ,h4 ,c4 ,f4 & ,l4 ,c4 ,m4 ( 7xy - y#b7c & y#c - x#by4 7,college ,entrance ,bo>d47 #a4 ,simplify 7a - #hm7_/7a7#a_/c7 - #bm7#a_/c77 - #b;a7#a_/c7m7#a_/c74 #b4 ,simplify1 writ+ ! result ) rational d5om9ator3 7a7#a_/b7 7#a7_/7x7-#a_/b77b - 7#a7_/7;a7-#a_/b77 - x7#a_/b7b7 _/ 7x a#b x#b7a_/b774 7,m4 ,i4 ,t47 #c4 ,f9d #g - #dh7a_/b77a_/b74 #d4 ,exp& 7a#c7a_/b7 - b#e7a_/b77e4 #e4 ,exp& & simplify 7#a - #bc7a_/b7 #cb7a_/b77b4 #f4 ,solve ! simultane\s equations x 7-#a_/b7 #by7-#a_/b7 #g_/f1 #bx7-#a_/b7 - y7-#a_/b7 #b_/c4 7,yale47 #g4 ,f9d to ?ree places ( decimals ! value ( 7a b77-#a_/c7_/7aa;a b#b77a_/f7 4 7a#c - b#c77-a_/b7_/7;a - b77#a_/b77a_/b71 :5 a #e & b #c4 7,columbia47 #h4 ,%[ t 7#aj - #de7a_/b77_/7e #ce7a_/b77 is ! negative ( ! reciprocal ( 7#aj #de7a_/b77_/7e - #ce7a_/b774 7,columbia47 #i4 ,solve & *eck #e_/cx #b7a_/b7 #cx #b7a_/b7 #cx - #a7a_/b74 #aj4 ,assum+ t :5 an apple falls f a tree ! di/ance 7,s met]s7 ?r\gh :i* x falls 9 any "t 7t seconds7 is giv5 by ! =mula ,s 7#a_/b7;gt#b 7:]e g #i4h71 f9d to two decimal places ! "t tak5 by an apple 9 fall+ #ae met]s4 7,college ,entrance ,bo>d47 ,excell5t practice may be obta9$ by solv+ ! ord9>y =mulas us$ 9 ari?metic1 geometry1 & physics orally1 = ea* lett] 9 turn4 ,,ari?metic p br i prt a p prt ,,geometry ,k 7#a_/b7 bh ,k bh ,k 7a#b7_/d #c7a_/b7 ,k 7#a_/b7 7b b'7 h ,k pi ,r#b ,c #b pi ,r ,k pi ,r ,l ,s #d pi ,r#b ,v pi ,r#b ,h ,v 7#a_/c7 pi ,r#b ,h ,v 7#d_/c7 pi ,r#c ,s 7pi ,r#b ,e7_/7#ahj7 ,c_/7,c'7 ,r_/7,r'7 ,k_/7,k'7 7,r#b7_/7,r'#b7 ,,physics v gt s 7#a_/b7 gt#b s 7v#b7_/7b;g7 ,c ,e_/,r ,e 7wv#b7_/7b;g7 e 7#d,pl#c7_/7;bh#c m7 ,e 7mv#b7_/7b7 t pi l_/g7#a_/b7 ,f 7m,v#b7_/7r7 mh 7mv#b7_/7b;g7 ,r gs_/7g s7 ,e 7#dn#bl#bw7_/7g7 ,c 7#e_/i77;,f - #cb7 ,,quadratic ,,equations #a4 ,def9e a quadratic equation2 a pure quadratic2 an affect$ 7or complete7 quadratic2 an equation 9 ! quadratic =m4 #b4 ,solve ! pure quadratic 7#g7_/7c,s#b7 - 7#aa7_/7i,s#b7 #e_/f4 ,review ! fir/ 7or usual7 me?od ( complet+ ! squ>e4 ,solve by x ! foll[+3 #c4 x#b #ajx #bd4 #d4 #bx#b - #ex #g4 #e4 7x - #a7_/b #b_/7x - #a7 #b-a_/b4 #f4 ax#b bx c #j4 ,review ! solution by factor+4 ,solve by x ! foll[+3 #g4 x#b #hx #g #j4 #h4 #bdx#b #bx #ae4 #i4 #c #ajx - #cx#b4 #aj4 -#g #fx - x#b4 ,solve1 by factor+1 ^! equations1 :i* are n quadratics3 #aa4 x#d #af4 #ab4 x#c #h4 #ac4 x#c x4 ,review ! solution by =mula4 ,solve by x ! foll[+3 #ad4 #ex#b - #fx #h4 #ae4 7#a_/b77x #a7 - 7x_/#c77bx - #a7 -#ab4 #af4 x#b #d;ax #ab;a#b4 #ag4 #cx#b #brx #br#b4 ,solve graphically3 #ah4 x#b - #bx - #h #j4 #ai4 x#b x - #b #j4 ,ref]5ce3 ,! *apt] on ,quadratic ,equations 9 any algebra 7fir/ "p ( ! *apt]74 #a4 ,solve by ?ree me?ods--=mula1 factor+1 & complet+ ! squ>e3 x#b #ajx #bd4 ,review equations 9 ! quadratic =m & solve3 #b4 x#d - #ex#b -#d4 #c4 #bx7-#b77a_/c7 - #cx7-#a77a_/c7 #b4 #d4 7x #c7_/7x - #c7 #f #e7x #c7_/7x - #c77a_/b74 7,let y 7x #c7_/7x - #c77a_/b7 & sub/itute47 #e4 #cx#b - #dx #bcx#b - #dx - #f7a_/b7 #ba4 #f4 x#b #ex - #e 7#f7_/7x#b #ex74 ,solve & *eck3 #g4 x #g7a_/b7 #cx - #b7a_/b7 7#dx #i7_/7cx - #b7a_/b774 #h4 x#b - #e7a_/b7 #f_/x#b - #e7a_/b7 #e4 #i4 7#ajw7_/7#ajw - #i7a_/b77 - #ajw #b7a_/b7 #b_/7ajw - #i7a_/b774 ,give results by inspection3 #aj4 7a7#a_/b7 b7#a_/b777;a7#a_/b7 - b7#a_/b774 #aa4 7#aj #ai7a_/b77a_/b777aj - #ai7a_/b77a_/b774 #ab4 ,h[ _m gallons ea* ( cream conta9+ #cc butt] fat & milk conta9+ #f butt] fat mu/ be mix$ to produce #aj gallons ( cream conta9+ #be butt] fat8 #ac4 ,i h #f 9 dimes1 qu>t]s1 & half-doll>s1 !re be+ #cc co9s 9 all4 ,! numb] ( dimes & qu>t]s toge!r is t5 times ! numb] ( half-doll>s4 ,h[ _m co9s ( ea* k9d are !re8 7,college ,entrance ,bo>d47 ,ref]5ce3 ,! la/ "p ( ! *apt] on ,quadratic ,equations 9 any algebra4 ,,! ,,!ory ,,( ,,quadratic ,,equations ,i4 ,to f9d ! sum & ! product ( ! roots4 ,! g5]al quadratic equation is ax#b bx c #j4 7#a7 ,or1 x#b 7b_/a7x c_/a #j4 7#b7 ,to d]ive ! =mula1 we h by transpos+ x#b 7b_/a7x -c_/a4 ,complet+ ! squ>e1 x#b 7b_/a7x b_/#b;a#b 7b#b7_/7d;a#b7 - c_/a 7b#b - #d;ac7_/7#d;a#b74 ,extract+ squ>e root1 x b_/#b;a -b#b - #d;ac7#a_/b7_/7b;a74 ,transpos+1 x -b_/#b;a - b#b - #d;ac7#a_/b7_/7b;a74 ,h5ce1 x -b - b#b - #d;ac7#a_/b7_/7b;a74 ,^! two values ( x we call roots4 ,= conv5i5ce repres5t !m by r#a & r#b4 ,h5ce1 r#a -b_/#b;a b#b - #d;ac7#a_/b7_/7b;a74 r#b -b_/#b;a - b#b - #d;ac7#a_/b7_/7b;a74 ---------------------------------------- ----- ,add+1 r#a r#b -7#b;b7_/7#b;a7 -b_/a4 7#c7 ,also1 r#a -b_/#b;a b#b - #d;ac7#a_/b7_/7b;a74 r#b -b_/#b;a - b#b - #d;ac7#a_/b7_/7b;a74 ---------------------------------------- --- ,multiply+1 r#a r#b 7b#b7_/7d;a#b7 - 7b#b - #d;ac7_/7#d;a#b7 7b#b - b#b #d;ac7_/7#d;a#b7 7#d;ac7_/7#d;a#b7 c_/a4 7#d7 ,h5ce we h %[n t r#a r#b -b_/a1 & r#a r#b c_/a4 ,or1 ref]r+ to equation 7#b7 above1 we h ! foll[+ rule3 ,:5 ! coeffici5t ( x#b is unity1 ! sum ( ! roots is ! coeffici5t ( x ) ! sign *ang$2 ! product ( ! roots is ! indep5d5t t]m4 ,,examples3 #a4 x#b - #ix #ba #j4 ,sum ( ! roots #i4 ,products ( ! roots #ba4 #b4 #cx#b - #gx - #ah #j4 ,sum ( ! roots #g_/c4 ,product ( ! roots -#f4 #c4 -#bax #ag - #dx#b4 ,sum ( ! roots #ba_/d4 ,product ( ! roots -#ag_/d4 ,,ii4 ,to f9d ! nature or "* ( ! roots4 ,z be=e1 r#a -b_/#b;a b#b - #d;ac7#a_/b7_/7b;a71 r#b -b_/#b;a - b#b - #d;ac7#a_/b7_/7b;a74 ,! b#b - #d;ac7#a_/b7 det]m9es ! nature or "* ( ! roots2 h5ce x is call$ ! discrim9ant4 ,if b#b - #d;ac is positive1 ! roots are real1 unequal1 & ei!r rational or irrational4 ,if b#b - #d;ac is negative1 ! roots are imag9>y & unequal4 ,if b#b - #d;ac is z]o1 ! roots are real1 equal1 & rational4 ,,examples3 #a4 x#b - #dx #b #j4 b#b - #d;ac7#a_/b7 #af - #h7a_/b7 #h7a_/b74 ,!re=e3 ,! roots are real1 unequal1 & irrational4 #b4 x#b - #dx #f #j4 b#b - #d;ac7#a_/b7 #af - #bd7a_/b7 -#h7a_/b74 ,!re=e3 ,! roots are imag9>y & unequal4 #c4 x#b - #dx #d #j4 b#b - #d;ac7#a_/b7 #af - #af7a_/b7 #j7a_/b74 ,!re=e3 ,! roots are real1 equal1 & rational4 ,,iii4 ,to =m ! quadratic equation :5 ! roots are giv54 ,suppose ! roots are #c1 -#g4 ,!n1 x #c1 ,or1 x - #c #j1 x -#g4 x #g #j4 ------------------- ,multiply+ to get a quadratic1 7x - #c77x #g7 #j4 ,or1 x#b #dx - #ba #j4 ,or1 use ! sum & product idea develop$ on ! prec$+ page4 ,! coeffici5t ( x#b mu/ be unity4 ,add ! roots & *ange ! sign to get ! coeffici5t ( x4 ,multiply ! roots to get ! indep5d5t t]m4 ,!re=e3 ,! equation is x#b #dx - #ba #j4 ,9 ! same way1 if ! roots are #b #c7a_/b7_/g1 #b - #c7a_/b7_/g1 ! equation is x#b - 7#d_/g7x #a_/di #j4 ,f9d ! sum1 ! product1 & ! nature or "* ( ! roots ( ! foll[+3 #a4 x#b - #gx #ab #j4 #b4 #ix#b - #fx #a #j4 #c4 x#b #bx #igcd #j4 #d4 #af #e_/x #ag_/7x#b74 #e4 7x - #h7_/7x - #c7 x4 #f4 7x #g77x - #f7 #gj4 #g4 x#b - x7#b77a_/b7 #c4 #h4 pr#b qr s #j4 ,=m ! equations ^: roots are3 #i4 #e1 -#c4 #aj4 #b_/c1 #e_/c4 #aa4 c d1 c - d4 #ab4 -#c1 -#e4 #ac4 #b - -#c7a_/b7_/e4 #ad4 #h_/c 7#b_/c7cg7a_/b71 #h_/c - 7#b_/c7cg7a_/b74 #ae4 -#b - -#b7a_/b7_/b4 #af4 ,solve x#b - #cx #d #j4 ,*eck by sub/itut+ ! values ( x2 !n *eck by f9d+ ! sum & ! product ( ! roots4 ,comp>e ! am\nt ( labor requir$ 9 ea* case4 #ag4 ,solve 7x - #c77x #b77x#b #cx - #d7 #j4 #ah4 ,is e7#dz7 #b;e7#cz7 e7#bz7 #b;ez #b e7-#bz7 a p]fect squ>e8 #ai4 ,f9d ! squ>e root 7%ort me?od73 7x#b - #a77x#b - #cx #b77x#b - x - #b74 #bj4 ,solve 7#a4bx - #a4e7_/7a4e7 74#dx #a7_/74bx - 4#b7 74#dx #a7_/74e74 #ba4 ,! glass ( a mirror is #ah in*es by #ab in*es1 & x has a frame ( uni=m wid? ^: area is equal to t ( ! glass4 ,f9d ! wid? ( ! frame4 ,,outl9e ,,( ,,simultane\s ,,quadratics ,simultane\s ,quadratics ,,case ,i4 ,"o equation l9e>4 ,! o!r quadratic4 #bx y #g1 x#b #by#b #bb4 ,,me?od3 ,solve = x z 9 t]ms ( y1 or vice v]sa1 9 ! l9e> & sub/itute 9 ! quadratic4 ,,case ,,ii4 ,bo? equations homog5e\s & ( ! second degree4 x#b - xy y#b #ci1 #bx#b - #cxy #by#b #dc4 ,,me?od3 ,let y vx1 & sub/itute 9 bo? equations4 ,,alt]nate ,,me?od3 ,solve = x 9 t]ms ( y 9 "o equation & sub/itute 9 ! o!r4 ,,case ,,iii4 ,any two ( ! quantities x y x#b y#b xy x - y x#c y#c x#c - y#c x#b xy y#b x#b - xy y#b giv54 x y #e1 x#b - xy y#b #g4 ,,me?od3 ,solve = x y & x - y2 !n add to get x1 subtract to get y4 ,,case ,,iv4 ,bo? equations symmetrical or symmetrical except = sign4 ,usually "o equation ( high degree1 ! o!r ( ! fir/ degree4 x#e y#e #bdb1 x y #b4 ,,me?od3 ,let x u v & y u - v1 & sub/itute 9 bo? equations4 ,special ,devices ,i4 ,consid] "s comp\nd quantity l xy1 x - y7#a_/b71 xy7#a_/b71 x_/y1 etc41 z ! unkn[n1 at fir/4 ,solve = ! comp\nd unkn[n1 & comb9e ! result+ equation ) ! simpl] orig9al equation4 x#b y#b xy #f1 x #by -#e4 ,,ii4 ,divide ! equations memb] by memb]4 ,!n solve by ,case ,i1 ,,ii1 or ,,iii4 x#c - y#c #aeb1 x - y #b4 ,,iii4 ,elim9ate ! quadratic t]ms4 ,!n solve by ,case ,i1 ,,ii1 or ,,iii4 xy x #ae1 xy y #af4 ,,simultane\s ,,quadratics ,solve3 #a4 x y #g1 x#b #dxy #eg4 #b4 #bx#b #df y#b1 xy y#b #ad4 #c4 x#b y#b #be1 x y #a4 #d4 x#d y#d #b1 x - y #b4 #e4 x#c y#c #bh1 x y #d4 #f4 x#b y#b xy - #ab #j1 x y #d4 #g4 #bxy - x #by #af1 #cxy #bx - #dy #aj4 #h4 7#cx - #by77#bx - #cy7 #bf1 x #a #by4 #i4 #dx#b #cxy #by#b #ah1 #cx#b #bxy - y#b #c4 #aj4 x#e y#e #bdb1 x y #b4 #aa4 x - y x - y7#a_/b7 #f1 xy #e4 #ab4 #dx#b - x y #fg1 #cx#b - #cy #bg4 #ac4 x - y - x - y7#a_/b7 #b1 x#c - y#c #bjdd4 7,yale47 #ad4 x#b xy x #ad1 y#b xy y #bh4 7,pr9ceton47 #ae4 x#b y#b #ac1 y#b #d7x - #b74 ,plot ! graph ( ea* equation4 7,cornell47 #af4 x#b y#b xy #cg1 x y xy - #ag4 7,columbia47 ,9 gr\p+ ! answ]s1 be sure to associate ea* value ( x ) ! correspond+ value ( y4 #ag4 ,! c\rse ( a ya*t is #cj miles 9 l5g? & is 9 ! %ape ( a "r triangle "o arm ( :i* is #b miles long] ?an ! o!r4 ,:at is ! di/ance along ea* side8 ,ref]5ce3 ,! *apt] on ,simultane\s ,quadratics 9 any algebra4 ,,ratio ,,& ,,proportion #a4 ,def9e ratio1 proportion1 mean proportional1 ?ird proportional1 f\r? proportional4 #b4 ,f9d a mean proportional betwe5 #d & #af2 #ah & #ej2 #abm#bn & #cmn#b4 #c4 ,f9d a ?ird proportional to #d & #g2 #e & #aj2 a#b - #i & a - #c4 #d4 ,f9d a f\r? proportional to #b1 #e1 & #d2 #ce1 #bj1 & #ad4 #e4 ,write out ! pro(s = ! foll[+1 /at+ ! !orem 9 full 9 ea* case3 7a7 ,! product ( ! extremes equals etc4 7b7 ,if ! product ( two numb]s equals ! product ( two o!r numb]s1 ei!r pair etc4 7c7 ,alt]nation4 7d7 ,inv]sion4 7e7 ,composition4 7f7 ,division4 7g7 ,composition & division4 7h7 ,9 a s]ies ( equal ratios1 ! sum ( ! antec$5ts is to ! sum ( ! consequ5ts etc4 7i7 ,l p[]s or l roots ( ! t]ms ( a proportion etc4 #f4 ,if x 3 m 33 #ac 3 #g1 write all ! possible proportions t c be d]iv$ f x4 ,see 7#e7 above4 #g4 ,giv5 rs #afam2 write ! eight proportions t may be d]iv$ f x1 & quote y\r au?ority4 #h4 7a7 ,:at !orem all[s y to *ange any proportion into an equation8 7b7 ,:at !orem all[s y to *ange any equation into a proportion8 #i4 ,if xy rg1 :at is ! ratio ( x to g8 ( y to r8 ( y to g8 #aj4 ,f9d two numb]s su* t _! sum1 diff]5ce1 & ! sum ( _! squ>es are 9 ! ratio #e 3 #c 3 #ea4 7,yale47 ,ref]5ce3 ,! *apt] on ,ratio & ,proportion 9 any algebra4 ,an easy & p[]ful me?od ( prov+ f\r expressions 9 proportion is illu/rat$ by ! foll[+ example3 ,giv5 a 3 b c 3 d2 prove t #c;a#c #e;ab#b 3 #c;a#c - #e;ab#b #c;c#c #e;cd#b 3 #c;c#c - #e;cd#b4 ,let a_/b r4 ,!re=e a br4 ,also c_/d r4 ,!re=e c dr4 ,sub/itute ! value ( a 9 ! fir/ ratio1 & c 9 ! second3 ,!n 7#c;a#c #e;ab#b7_/7c;a#c - #e;ab#b7 7#c;b#cr#c #e;b#cr7_/7#c;b#cr#c - #e;b#cr7 b#cr7#cr#b #e7_/;b#cr7#cr#b - #e7 7#cr#b #e7_/7cr#b - #e74 ,also 7#c;c#c #e;cd#b7_/7c;c#c - #e;cd#b7 7#c;d#cr#c #e;d#cr7_/7#c;d#cr#c - #e;d#cr7 d#cr7#cr#b #e7_/;d#cr7#cr#b - #e7 7#cr#b #e7_/7cr#b - #e74 ,!re=e 7#c;a#c #e;ab#b7_/7c;a#c - #e;ab#b7 7#c;c#c #e;cd#b7_/7c;c#c - #e;cd#b74 ,axiom #a4 ,or1 #c;a#c #e;ab#b 3 #c;a#c - #e;ab#b #c;c#c #e;cd#b 3 #c;c#c - #e;cd#b4 ,if a 3 b c 3 d1 prove3 #a4 a#b b#b 3 a#b c#b d#b 3 c#b4 #b4 a#b #c;b#b 3 a#b - #c;b#b c#b #c;d#b 3 c#b - #c;d#b4 #c4 a#b #b;b#b 3 #b;b#b ac #b;bd 3 #b;bd4 #d4 #b;a #c;c 3 #b;a - #c;c #h;b #ab;d 3 #h;b - #ab;d4 #e4 a#b - ab b#b 3 7a#c - b#c7_/;a c#b - cd d#b 3 7c#c - d#c7_/;c4 #f4 ,! second ( ?ree numb]s is a mean proportional betwe5 ! o!r two4 ,! ?ird numb] exce$s ! sum ( ! o!r two by #bj2 & ! sum ( ! fir/ & ?ird exce$s ?ree times ! second by #d4 ,f9d ! numb]s4 #g4 ,?ree numb]s are proportional to #e1 #g1 & #i2 & _! sum is #ad4 ,f9d ! numb]s4 7,college ,entrance ,bo>d47 #h4 ,a triangul> field has ! sides #ae1 #ah1 & #bg rods1 respectively4 ,f9d ! dim5sions ( a simil> field hav+ #d times ! area4 ,,ari?metical ,,progression #a4 ,def9e an ari?metical progression4 ,le>n to d]ive ! ?ree =mulas 9 ari?metical progression3 l a 7n - #a7;d1 ,s 7n_/#b77;a l71 ,s 7n_/#b7b;a 7n - #a7;d4 #b4 ,f9d ! sum ( ! fir/ #ej odd numb]s4 #c4 ,9 ! s]ies #b1 #e1 #h1 4441 :i* t]m is #ib8 #d4 ,h[ _m t]ms mu/ be tak5 f ! s]ies #c1 #e1 #g1 4441 to make a total ( #bee8 #e4 ,ins]t #e ari?metical means betwe5 #aa & #cb4 #f4 ,ins]t #i ari?metical means betwe5 #g-a_/b & #cj4 #g4 ,f9d x1 if #c #bx1 #e #fx1 #i #ex are 9 ,a4 ,p4 #h4 ,! #g? t]m ( an ari?metical progression is #ag1 & ! #ac? t]m is #ei4 ,f9d ! #d? t]m4 #i4 ,h[ c y turn an ,a4 ,p4 into an equation8 #aj4 ,giv5 a -#e_/c1 n #bj1 ,s -#e_/c1 f9d d & l4 #aa4 ,f9d ! sum ( ! fir/ n odd numb]s4 #ab4 ,an ari?metical progression consi/s ( #ba t]ms4 ,! sum ( ! ?ree t]ms 9 ! middle is #abi2 ! sum ( ! la/ ?ree t]ms is #bcg4 ,f9d ! s]ies4 7,look up ! %ort me?od = su* problems47 7,mass4 ,in/4 ( ,te*nology47 #ac4 ,b travels #c miles ! fir/ day1 #g miles ! second day1 #aa miles ! ?ird day1 etc4 ,9 h[ _m days w ,b ov]take ,a :o />t$ f ! same po9t #h days 9 advance & :o travels uni=mly #ae miles a day8 ,ref]5ce3 ,! *apt] on ,ari?metical ,progression 9 any algebra4 ,,geometrical ,,progression #a4 ,def9e a geometrical progression4 ,le>n to d]ive ! f\r =mulas 9 geometrical progression3 ,i4 l ar7n - #a74 ,,ii4 ,s 7arn - a7_/7r - #a74 ,,iii4 ,s 7rl - a7_/7r - #a74 ,,iv4 ,sinf9ity 7a7_/7#a - r74 #b4 ,h[ _m t]ms mu/ be tak5 f ! s]ies #i1 #ah1 #cf1 444 to make a total ( #efg8 #c4 ,9 ! ,g4 ,p4 #b1 #f1 #ah1 4441 :i* t]m is #dhf8 #d4 ,f9d x1 if #bx - #d1 #ex - #g1 #ajx #d are 9 geometrical progression4 #e4 ,h[ c y turn a ,g4 ,p4 into an equation8 #f4 ,ins]t #d geometrical means betwe5 #d & #igb4 #g4 ,ins]t #f geometrical means betwe5 #e_/af & #eabj4 #h4 ,giv5 a -#b1 n #e1 l -#cb2 f9d r & ,s4 #i4 ,if ! fir/ t]m ( a geometrical progression is #ab & ! sum to inf9ity is #cf1 f9d ! #d? t]m4 #aj4 ,if ! s]ies #c-a_/c1 #b-a_/b1 444 be an ,a4 ,p41 f9d ! #ig? t]m4 ,if a ,g4 ,p41 f9d ! sum to inf9ity4 #aa4 ,! ?ird t]m ( a geometrical progression is #cf2 ! #f? t]m is #igb4 ,f9d ! fir/ & second t]ms4 #ab4 ,ins]t betwe5 #f & #af two numb]s1 su* t ! fir/ ?ree ( ! f\r %all be 9 ari?metical progression1 & ! la/ ?ree 9 geometrical progression4 #ac4 ,a rubb] ball falls f a height ( #dj in*es & on ea* reb\nd rises #dj ( ! previ\s height4 ,f9d by =mula h[ f> x falls on its eigh? desc5t4 7,yale47 ,ref]5ce3 ,! *apt] on ,geometrical ,progression 9 any algebra4 ,,! ,,b9omial ,,!orem #a4 ,review ! ,b9omial ,!orem laws4 7,see ,involution47 ,exp&3 #b4 7b - n7#g4 #c4 7x x7-#a77e4 #d4 a_/x - x_/a#f4 #e4 x_/#by - xy7#a_/b7e4 #f4 7x#b - x #b7c4 #g4 7#b;b#b7a_/c77_/7y7 7#cy7#a_/b77_/7;b#c7d4 #h4 7a b7n an na7n - #a7;b n7n - #a7_/7a4b7 a7n - #b7;b#b n7n - #a77n - #b7_/7a4b4c7 a7n - #c7;b#c n7n - #a77n - #b77n - #c7_/7a4b4c4d7 a7n - #d7 b#d 4444 ,%[ by obs]vation t ! =mula = ! 7r #a7? t]m n7n - #a77n - #b74447n - r #a7_/a4b4c4d 444 r a7n - r7br4 #i4 ,indicate :at ! #ig? t]m ( 7a b7n w\ld be4 #aj4 ,us+ ! expansion ( 7a b7n 9 7#h71 d]ive a =mula = ! r? t]m by obs]v+ h[ ea* t]m is made up1 !n g5]aliz+4 ,us+ ei!r ! =mula 9 7#h7 or 7#aj71 :i*ev] y are famili> )1 f9d3 #aa4 ,! #d? t]m ( a #a_/;a7#cj74 #ab4 ,! #h? t]m ( 7#a xy7#a_/b777ac74 #ac4 ,! middle t]m ( 7#b;a7#c_/d7 - ya7#a_/c777aj74 #ad4 ,! t]m n conta9+ x 9 x#c - #b_/x7#ab74 #ae4 ,! t]m conta9+ x7#ah7 9 x#b - a_/x7#ae74 ,ref]5ce3 ,! *apt] on ,! ,b9omial ,!orem 9 any algebra4 ,,miscellane\s ,,examples1 ,,quadratics ,,& ,,beyond #a4 ,solve ! equation x#b - #a4fx - 4#bc #j1 obta9+ ! values ( ! roots correct to ?ree significant figures4 7,h>v>d47 #b4 ,write ! roots ( 7x#b #bx77x#b - #bx - #c77x#b - x #a7 #j4 7,%effield ,sci5tific ,s*ool47 #c4 ,solve #bbx #b7a_/b7 #bx #a7a_/b7 7#abx #d7_/7hx #ha_/b74 7,yale47 #d4 ,solve ! equation ,v 7,h_/#c77;,b x ,bx7#a_/b77 = x1 tak+ ,h #f1 ,b #h1 & ,v #bh2 & v]ify y\r result4 7,h>v>d47 #e4 ,solve x 3 y #b 3 #c1 x#b y#b #e7x y7 #b4 #f4 ,solve #bx#b - #dx #cx#b - #bx #f7a_/b7 #ae4 7,coll4 ,ent4 ,bo>d47 #g4 ,f9d all values ( x & y :i* satisfy ! equations3 x7#a_/b7 y7#a_/b7 #d1 #a_/x #a7a_/b7 - x7#a_/b7 - #a_/x #a7a_/b7 x7#a_/b7 y4 7,mass4 ,in/4 ( ,te*nology47 #h4 ,if alpha & beta repres5t ! roots ( px#b qx r #j1 f9d alpha beta1 alpha - beta1 & alphabeta 9 t]ms ( p1 q1 & r4 7,pr9ceton47 #i4 ,=m ! equation ^: roots are #b #c7a_/b7 & #b - -#c7a_/b74 #aj4 ,det]m9e1 )\t solv+1 ! "* ( ! roots ( #ix#b - #bdx #af #j4 7,college ,entrance ,bo>d47 #aa4 ,if a 3 b c 3 d1 prove t a b 3 c d a#b b#b7a_/b7 3 c#b d#b7a_/b74 7,college ,entrance ,bo>d47 #ab4 ,giv5 a 3 b c 3 d4 ,prove t a#b b#b 3 7a#c7_/7;a b7 c#b d#b 3 7c#c7_/7;c d74 7,%effield47 #ac4 ,! #i? t]m ( an ari?metical progression is #a_/f2 ! #af? t]m is #e_/b4 ,f9d ! fir/ t]m4 7,reg5ts47 ,solve graphically3 #a4 x#b - x - #f #j4 #b4 x#b #cx - #aj #j4 #c4 ,f9d f\r numb]s 9 ari?metical progression1 su* t ! sum ( ! fir/ two is #a1 & ! sum ( ! la/ two is -#ai4 #d4 ,:at numb] add$ to #b1 #bj1 #i1 #cd1 w make ! results proportional8 #e4 ,f9d ! middle t]m ( #c;a#e 7b7#c_/d77_/7b7h4 #f4 ,solve 7x #a7_/7cx #b7 7#bx - #c7_/7cx - #b7 - #a - #cf_/7d - #ix#b74 7,pr9ceton47 #g4 ,a /rip ( c>pet "o half in* ?ick & #bi-f_/g feet long is roll$ on a roll] f\r in*es 9 diamet]4 ,f9d h[ _m turns !re w be1 rememb]+ t ea* turn increases ! diamet] by "o in*1 & t ! circumf]5ce ( a circle equals 7approximately7 #bb_/g times ! diamet]4 7,h>v>d47 #h4 ,! sum ( ! fir/ ?ree t]ms ( a geometrical progression is #ba1 & ! sum ( _! squ>es is #ahi4 ,:at is ! fir/ t]m8 7,yale47 #i4 ,f9d ! geometrical progression ^: sum to inf9ity is #d1 & ^: second t]m is #c_/d4 #aj4 ,solve #dx #dcx#b - #gx #c7a_/b7 #cx#b - #cx #f4 #aa4 ,solve #bx#b #cxy - #ey#b #d1 #bxy #cy#b -#c4 #ab4 ,two hundr$ /ones are plac$ on ! gr\nd #c feet ap>t1 ! fir/ be+ #c feet f a basket4 ,if ! basket & all ! /ones are 9 a /raight l9e1 h[ f> does a p]son travel :o />ts f ! basket & br+s ! /ones to x "o by "o8 ,solve graphically2 & *eck by solv+ algebraically3 #a4 x#b y#b #be1 x y #a4 #b4 x#b - #cx - #ah #j4 #c4 x#b #cx - #aj #j4 ,det]m9e ! value ( m = :i* ! roots ( ! equation w be equal3 7,,h9t3 ,see page #dj4 ,to h ! roots equal1 b#b - #d;ac mu/ equal #j47 #d4 #bx#b - mx #ab-a_/b #j4 #e4 7m - #a7x#b mx #bm - #c #j4 #f4 ,if #b;a #c;b is a root ( x#b - #f;bx - #d;a#b #i;b#b #j1 f9d ! o!r root )\t solv+ ! equation4 7,univ4 ( ,p5n47 #g4 ,h[ _m times does a common clock /rike 9 #ab h\rs8 #h4 ,f9d ! sum to inf9ity ( #b_/7b7a_/b771 #a_/7b7a_/b771 #a_/7bb7a_/b771 4444 #i4 ,solve x_/#b #f_/x#b - #fx_/#b #f_/x #h #j4 #aj4 ,f9d ! value ( ! recurr+ decimal #b4badbad4444 #aa4 ,a man pur*ases a #ejj piano by pay+ mon?ly in/allm5ts ( #aj & int]e/ on ! debt4 ,if ! ye>ly rate is #f1 :at is ! total am\nt ( int]e/8 #ab4 ,! ari?metical mean betwe5 two numb]s is #db-a_/b1 & _! geometrical mean is #db4 ,f9d ! numb]s4 7,college ,entrance ,exam4 ,bo>d47 #ac4 ,if ! middle t]m ( #cx - 7#a7_/7bx7#a_/b77d is equal to ! f\r? t]m ( #bx7#a_/b7 #a_/bx#g1 f9d ! value ( x4 7,m4 ,i4 ,t47 ,,problems ,l9e> ,equations1 ,"o ,unkn[n #a4 ,a tra9 runn+ #cj miles an h\r requires #ba m9utes long] to g a c]ta9 di/ance ?an does a tra9 runn+ #cf miles an h\r4 ,h[ great is ! di/ance8 7,cornell47 #b4 ,a man c walk #b-a_/b miles an h\r up hill & #c-a_/b miles an h\r d[n hill4 ,he walks #ef miles 9 #bj h\rs on a road no "p ( :i* is level4 ,h[ mu* ( x is up hill8 7,yale47 #c4 ,a physician hav+ #ajj cubic c5timet]s ( a #f solution ( a c]ta9 m$ic9e wi%es to dilute x to a #c-a_/b solution4 ,h[ mu* wat] mu/ he add8 7,a #f solution conta9s #f ( m$ic9e & #id ( wat]47 7,case47 #d4 ,a cl]k e>n$ #ejd 9 a c]ta9 numb] ( mon?s4 ,8 sal>y 0 increas$ #be1 & he !n e>n$ #dej 9 two mon?s less "t ?an x _h previ\sly tak5 him to e>n #ejd4 ,:at 0 8 orig9al sal>y p] mon?8 7,college ,entrance ,bo>d47 #e4 ,a p]son :o possesses #ae1jjj employs a "p ( ! money 9 build+ a h\se4 ,he inve/s "o ?ird ( ! money :i* rema9s at #f1 & ! o!r two ?irds at #i1 & f ^! inve/m5ts he obta9s an annual income ( #ejj4 ,:at 0 ! co/ ( ! h\se8 7,m4 ,i4 ,t47 #f4 ,two travel]s h toge!r #djj p\nds ( baggage4 ,"o pays #a4bj & ! o!r #a4hj = excess above ! weight c>ri$ free4 ,if all _h belong$ to "o p]son1 he w\ld h _h to pay #d4ej4 ,h[ mu* baggage is all[$ to g free8 7,yale47 #g4 ,a man :o c r[ #d-a_/c miles an h\r 9 /ill wat] r[s d[n/ream & returns4 ,! rate ( ! curr5t is #b-a_/d miles p] h\r1 & ! "t requir$ = ! trip is #ac h\rs4 ,h[ _m h\rs does he require to return8 ,simultane\s ,equations1 ,two & ,?ree ,unkn[ns #a4 ,a manual tra9+ /ud5t 9 mak+ a bookcase f9ds t ! di/ance f ! top ( ! l[e/ %elf to ! "u side ( ! top %elf is #d ft4 #f 94 ,he desires to put betwe5 ^! f\r o!r %elves ( in* bo>ds 9 su* a way t ! book space w dim9i% "o in* = ea* %elf f ! bottom to ! top4 ,:at w be ! sev]al spaces betwe5 ! %elves8 #b4 ,a quantity ( wat]1 suffici5t to fill ?ree j>s ( diff]5t sizes1 w fill ! smalle/ j> #d times1 or ! l>ge/ j> twice ) #d gallons to sp>e1 or ! second j> ?ree times ) #b gallons to sp>e4 ,:at is ! capacity ( ea* j>8 7,case47 #c4 ,a policeman is *as+ a pickpocket4 ,:5 ! policeman is #hj y>ds beh9d him1 ! pickpocket turns up an alley2 b com+ to ! end1 he f9ds !re is no outlet1 turns back1 & is caught j z he comes out ( ! alley4 ,if he _h discov]$ t ! alley _h no outlet :5 he _h run halfway up & _h !n turn$ back1 ! policeman w\ld h _h to pursue ! ?ief #abj y>ds beyond ! alley be=e cat*+ him4 ,h[ long is ! alley8 7,h>v>d47 #d4 ,a & ,b toge!r c d a piece ( "w 9 #ad days4 ,aft] !y h work$ #f days on x1 !y are jo9$ by ,c :o works twice z fa/ z ,a4 ,! ?ree f9i% ! "w 9 #d days4 ,h[ long w\ld x take ea* man alone to d x8 7,columbia47 #e4 ,9 a c]ta9 mill "s ( ! workm5 receive #a4ej a day1 o!rs m4 ,! total paid 9 wages ea* day is #cej4 ,an assessm5t made by a labor union to raise #bjj requires #a4jj f ea* man receiv+ #a4ej a day1 & half ( "o day's pay f e man receiv+ m4 ,h[ _m m5 receive #a4ej a day8 7,h>v>d47 #f4 ,!re are two alloys ( silv] & copp]1 ( :i* "o conta9s twice z mu* copp] z silv]1 & ! o!r ?ree times z mu* silv] z copp]4 ,h[ mu* mu/ be tak5 f ea* to obta9 a kilogram ( an alloy to conta9 equal quantities ( silv] & copp]8 7,m4 ,i4 ,t47 #g4 ,two automobiles travel t[>d ea* o!r ov] a di/ance ( #abj miles4 ,a leaves at #i ,a4,m41 #a h\r be=e ,b />ts to meet him1 & !y meet at #ab3jj ,m4 ,if ea* _h />t$ at #i3ae ,a4,m41 !y w\ld h met at #ab3jj ,m4 also4 ,f9d ! rate at :i* ea* travel$4 7,m4 ,i4 ,t47 ,quadratic ,equations #a4 ,telegraph poles are set at equal di/ances ap>t4 ,9 ord] to h two less to ! mile1 x w be necess>y to set !m #bj feet f>!r ap>t4 ,f9d h[ f> ap>t !y are n[4 7,yale47 #b4 ,! di/ance ,s t a body falls f re/ 9 t seconds is giv5 by ! =mula ,s #aft#b4 ,a man drops a /one into a well & he>s ! spla% aft] #c seconds4 ,if ! velocity ( s\nd 9 air is #ajhf feet a second1 :at is ! dep? ( ! well8 7,yale47 #c4 ,x requires #bjjj squ>e tiles ( a c]ta9 size to pave a hall1 or #cabe squ>e tiles ^: dim5sions are "o in* less4 ,f9d ! area ( ! hall4 ,h[ _m solutions has ! equation ( ?is problem8 ,h[ _m has ! problem itself8 ,expla9 ! app>5t discrepancy4 7,cornell47 #d4 ,a rectangul> tract ( l&1 #hjj feet long by #fjj feet broad1 is divid$ into f\r rectangul> blocks by two /reets ( equal wid? runn+ ?r\gh x at "r angles4 ,f9d ! wid? ( ! /reets1 if toge!r !y cov] an area ( #gg1ejj squ>e feet4 7,m4 ,i4 ,t47 #e4 7a7 ,! height y to :i* a ball ?r[n v]tically upw>d ) a velocity ( #ajj feet p] second rises 9 x seconds is giv5 by ! =mula1 y #ajjx - #afx#b4 ,9 h[ _m seconds w ! ball rise to a height ( #add feet8 7b7 ,draw ! graph ( ! equation y #ajjx - #afx#b4 7,college ,entrance ,bo>d47 #f4 ,two laun*es race ov] a c\rse ( #ab miles4 ,! fir/ /eams #g-a_/b miles an h\r4 ,! o!r has a />t ( #aj m9utes1 runs ov] ! fir/ half ( ! c\rse ) a c]ta9 spe$1 b increases its spe$ ov] ! second half ( ! c\rse by #b miles p] h\r1 w9n+ ! race by a m9ute4 ,:at is ! spe$ ( ! second laun*8 ,expla9 ! mean+ ( ! negative answ]4 7,%effield ,sci5tific ,s*ool47 #g4 ,! circumf]5ce ( a re> :eel ( a c]ta9 wagon is #c feet m ?an ! circumf]5ce ( a front :eel4 ,! re> :eel p]=ms #ajj few] revolutions ?an ! front :eel 9 travel+ a di/ance ( #fjjj feet4 ,h[ l>ge are ! :eels8 7,h>v>d47 #h4 ,a man />ts f home to cat* a tra91 walk+ at ! rate ( #a y>d 9 #a second1 & arrives #b m9utes late4 ,if he _h walk$ at ! rate ( #d y>ds 9 #c seconds1 he w\ld h arriv$ #b-a_/b m9utes e>ly4 ,f9d ! di/ance f 8 home to ! /ation4 7,college ,entrance ,bo>d47 ,simultane\s ,quadratics #a4 ,two cubical coal b9s toge!r hold #bhj cubic feet ( coal1 & ! sum ( _! l5g?s is #aj feet4 ,f9d ! l5g? ( ea* b94 #b4 ,! sum ( ! radii ( two circles is #be in*es1 & ! diff]5ce ( _! areas is #abepi squ>e in*es4 ,f9d ! radii4 #c4 ,! area ( a "r triangle is #aej squ>e feet1 & its hypot5use is #be feet4 ,f9d ! arms ( ! triangle4 #d4 ,! comb9$ capacity ( two cubical tanks is #fcg cubic feet1 & ! sum ( an edge ( "o & an edge ( ! o!r is #ac feet4 7a7 ,f9d ! l5g? ( a diagonal ( any face ( ea* cube4 7b7 ,f9d ! di/ance f upp] left-h& corn] to l[] "r-h& corn] 9 ei!r cube4 #e4 ,a & ,b run a mile4 ,9 ! fir/ heat ,a gives ,b a />t ( #bj y>ds & beats him by #cj seconds4 ,9 ! second heat ,a gives ,b a />t ( #cb seconds & beats him by #i-e_/aa y>ds4 ,f9d ! rate at :i* ea* runs4 7,%effield47 #f4 ,aft] /reet improvem5t x is f\nd t a c]ta9 corn] rectangul> lot has lo/ #a_/aj ( its l5g? & #a_/ae ( its wid?4 ,its p]imet] has be5 decreas$ by #bh feet1 & ! new area is #cjbd squ>e feet4 ,f9d ! r$uc$ dim5sions ( ! lot4 7,college ,entrance ,bo>d47 #g4 ,a man sp5ds #eci = %eep4 ,he keeps #ad ( ! flock t he buys1 & sells ! rema9d] at an advance ( #b p] head1 ga9+ #bh by ! transaction4 ,h[ _m %eep did he buy1 & :at 0 ! co/ ( ea*8 7,yale47 #h4 ,a boat's crew1 r[+ at half _! usual spe$1 r[ #c miles d[n/ream & back aga9 9 #b h\rs & #dj m9utes4 ,at full spe$ !y c g ov] ! same c\rse 9 #a h\r & #d m9utes4 ,f9d ! rate ( ! crew1 & ! rate ( ! curr5t 9 miles p] h\r4 7,college ,entrance ,bo>d47 #i4 ,f9d ! sides ( a rectangle ^: area is un*ang$ if its l5g? is increas$ by #d feet & its bread? decreas$ by #c feet1 b :i* loses "o ?ird ( its area if ! l5g? is increas$ by #af feet & ! bread? decreas$ by #aj feet4 7,m4 ,i4 ,t47 ,,college ,,entrance ,,exam9ations ,,univ]sity ,,( ,,cali=nia ,,elem5t>y ,,algebra #a4 ,if a #d1 b -#c1 c #b1 & d -#d1 f9d ! value (3 7a7 ab#c - #c;cd#b #b7c;a - b77c - #b;d74 7b7 #b;a#c - #c;b#d 7#d;c#c d#c77d;c#b d#b74 #b4 ,r$uce to a mix$ numb]3 7#c;a#d - #d;a#c - #aj;a#b #da;a - #bh7_/7;a#b - #c;a #d74 ,simplify3 #c4 7a #b7_/7;a#b #c;a - #dj7 - 7b - #b7_/7;ab - #e;b #c;a - #ae74 #d4 #a - 7#b - #c;b - #b;c7_/7a #b7 _/ 7a#b - #d;c#b #i;b#b #f;ab7_/7#b;a#b a - #f74 #e4 ,a's age #aj ye>s h5ce w be #d times :at ,b's age 0 #aa ye>s ago1 & ! am\nt t ,a's age exce$s ,b's age is "o ?ird ( ! sum ( _! ages #h ye>s ago4 ,f9d _! pres5t ages4 #f4 ,draw ! l9es repres5t$ by ! equations #cx - #by #ac & #bx #ey -#d1 & f9d by algebra ! coo]d9ates ( ! po9t :]e !y int]sect4 #g4 ,solve ! equations bx - ay b#b - ab1 y - b #b7x - #b;a74 #h4 ,solve 7#bx #a77cx - #b7 - 7#ex - #g77x - #b7 #da4 ,,colorado ,,s*ool ,,( ,,m9es ,,elem5t>y ,,algebra #a4 ,solve by factor+3 x#c #cjx #aax#b4 #b4 ,%[ t #a - 7a#b b#b - c#b7_/7b;ab7#b 7a b c77a b - c77a - b c77b c - a7 _/ #d;a#b;b#b4 #c4 ,h[ _m pairs ( numb]s w satisfy simultane\sly ! two equations #cx #by #g1 x y #c8 ,%[ by means ( a graph t y\r answ] is correct4 ,:at is meant by elim9at+ x 9 ! above equations by sub/itution8 by comp>ison8 by subtraction8 #d4 ,f9d ! squ>e root ( #bbc1gbh4 #e4 ,simplify3 7a7 #a_/c7a_/b7 #ab7a_/b7 - #c_/d7a_/b74 7b7 7--#c-d7a_/b77a_/b77d4 #f4 ,solve ! equation 4#jcx#b - #b4bcx #a4ajge #j4 #g4 ,h[ f> mu/ a boy run 9 a potato race if !re are n potatoes 9 a /raight l9e at a di/ance d feet ap>t1 ! fir/ be+ at a di/ance a feet f ! basket8 ,,columbia ,,univ]sity ,,elem5t>y ,,algebra ,,complete ,,"t3 ,,?ree ,,h\rs ,six que/ions are requir$2 two f ,gr\p ,a1 two f ,gr\p ,b1 & bo? que/ions ( ,gr\p ,c4 ,no extra cr$it w be giv5 = m ?an six que/ions4 ,gr\p ,a #a4 7a7 ,resolve ! foll[+ into _! prime factors3 7#a7 7x#b - y#b7b - y#d4 7#b7 #ajx#b - #gx - #f4 7b7 ,f9d ! ,h4 ,c4 ,f4 & ! ,l4 ,c4 ,m4 ( x#c - #cx#b x - #c1 x#c - #cx#b - x #c4 #b4 7a7 ,simplify x_/y y_/x - #b_/a_/x #a_/y x_/y y_/x #b_/a_/x - #a_/y4 7b7 ,if x 3 y 7x - z7#b 3 7y - z7#b1 prove t z is a mean proportional betwe5 x & y4 #c4 ,a crew c r[ #aj miles 9 #ej m9utes d[n/ream1 & #ab miles 9 an h\r & a half up/ream4 ,f9d ! rate ( ! curr5t & ( ! crew 9 /ill wat]4 ,gr\p ,b #d4 7a7 ,det]m9e ! values ( k s t ! equation 7#b k7x#b #bkx #a #j %all h equal roots4 7b7 ,solve ! equations x#b - xy y#b #g1 #bx - #cy #j4 7c7 ,plot ! foll[+ two equations1 & f9d f ! graphs ! approximate values ( _! common solutions3 x#b y#b #be1 #dx#b #iy#b #add4 #e4 ,two integ]s are 9 ! ratio #d 3 #e4 ,increase ea* by #ae1 & ! diff]5ce ( _! squ>es is #iii4 ,:at are ! integ]s8 #f4 ,a man has #eci to sp5d = %eep4 ,he wi%es to keep #ad ( ! flock t he buys1 b to sell ! rema9d] at a ga9 ( #b p] head4 ,?is he does & ga9s #bh4 ,h[ _m %eep did he buy1 & at :at price ea*8 ,gr\p ,c #g4 7a7 ,f9d ! sev5? t]m ( a #a_/;a7#ac74 7b7 ,d]ive ! =mula = ! sum ( n t]ms ( an ari?metic progression4 #h4 ,a ball fall+ f a height ( #fj feet reb\nds aft] ea* fall "o ?ird ( its la/ desc5t4 ,:at di/ance has x pass$ ov] :5 x /rikes ! gr\nd = ! eigh? "t8 ,,cornell ,,univ]sity ,,elem5t>y ,,algebra #a4 ,f9d ! ,h4 ,c4 ,f43 x#d - y#d1 x#c - xy#b x#by - y#c1 x#d #bx#by#b - #cy#d4 #b4 ,solve ! foll[+ set ( equations3 x y -#a1 x #cy #bz -#d1 x - y #dz #e4 #c4 ,exp& & simplify3 #bx#c - #a_/x#g4 #d4 ,an automobile goes #hj miles & back 9 #i h\rs4 ,! rate ( spe$ return+ 0 #d miles p] h\r fa/] ?an ! rate go+4 ,f9d ! rate ea* way4 #e4 ,simplify3 7x #a7_/7x - #a7b - #b 7x - #a7_/7x #a7b _/7x #a7_/7x - #a7b - 7x - #a7_/7x #a7b4 #f4 ,solve = x3 7#bx #c7_/7x - #a7 - #f #e_/7x#b #bx - #c74 #g4 ,a1 ,b1 & ,c1 all work+ toge!r1 c d a piece ( "w 9 #b-b_/c days4 ,a works twice z fa/ z ,c1 & ,a & ,c toge!r c\ld d ! "w 9 #d days4 ,h[ long w\ld x take ea* "o ( ! ?ree to d ! "w alone8 ,,cornell ,,univ]sity ,,int]m$iate ,,algebra #a4 ,solve ! foll[+ set ( equations3 x y -#a1 #bz #ew #a1 x #cy #bz -#d1 x - y #dz #dw #e4 #b4 ,simplify3 7a7 #f - #bj7a_/b77a_/b74 7b7 #a x#b #a7a_/b7_/a x#b #a7a_/b7 x#b4 #c4 ,f9d1 & simplify1 ! #bc;d t]m 9 ! expansion ( 7#bx#b7_/7c7 - #c_/d7bh74 #d4 ,! weight ( an object v>ies directly z its di/ance f ! c5t] ( ! e>? :5 x is bel[ ! e>?'s surface1 & inv]sely z ! squ>e ( its di/ance f ! c5t] :5 x is above ! surface4 ,if an object weighs #aj p\nds at ! surface1 h[ f> above1 & h[ f> bel[ ! surface w x weigh #i p\nds8 7,! radius ( ! e>? may be tak5 z #djjj miles47 #e4 ,solve ! foll[+ pair ( equations = x & y3 x#b y#b #d1 x 7#a #b7a_/b77y - #b4 #f4 ,f9d ! value ( #a #h7-x_/#c7_/7hx77#a_/b7 #aj7x - #b71 :5 x #b4 #g4 ,f a squ>e ( pa/ebo>d1 #ab in*es on a side1 squ>e corn]s are cut1 & ! sides are turn$ up to =m a rectangul> box4 ,if ! squ>es cut out f ! corn]s _h be5 #a in* l>g] on a side1 ! volume ( ! box w\ld h be5 increas$ #bh cubic in*es4 ,:at is ! size ( ! squ>e corn]s cut out8 7,see ! figure on ! blackbo>d47 ,,h>v>d ,,univ]sity ,,elem5t>y ,,algebra ,,"t3 ,,"o ,,h\r ,,& ,a ,,half ,arrange y\r "w neatly & cle>ly1 beg9n+ ea* "q on a sep>ate page4 #a4 ,simplify ! foll[+ expression3 #a_/;a #a_/7;b c7_/#a_/;a - #a_/7;b c7 #a 7b#b c#b - a#b7_/7b;bc74 #b4 7a7 ,write ! middle t]m ( ! expansion ( 7a - b7#ad by ! b9omial !orem4 7b7 ,f9d ! value ( a#g;b#g1 if a x7#b_/g7y7-#c_/b7 & b 7#a_/b7 x7-#a_/g7y7#a_/b71 & r$uce ! result to a =m hav+ only positive expon5ts4 #c4 ,f9d correct to ?ree significant figures ! negative root ( ! equation #a - #b_/7x #a7 #dx_/7x #a7b #j4 #d4 ,prove ! rule = f9d+ ! sum ( n t]ms ( a geometrical progression ( :i* ! fir/ t]m is a & ! con/ant ratio is r4 ,f9d ! sum ( #h t]ms ( ! progression #e #c-a_/c #b-b_/i 4444 #e4 ,a goldsmi? has two alloys ( gold1 ! fir/ be+ #c_/d pure gold1 ! second #e_/ab pure gold4 ,h[ mu* ( ea* mu/ he take to produce #ajj ounces ( an alloy :i* %all be #b_/c pure gold8 ,,h>v>d ,,univ]sity ,,elem5t>y ,,algebra ,,"t3 ,,"o ,,h\r ,,& ,a ,,half #a4 ,solve ! simultane\s equations x y a b1 7y b7_/7x a7 a_/b1 & v]ify y\r results4 #b4 ,solve ! equation x#b - #a4fx - #j4bc #j1 obta9+ ! values ( ! roots correct to ?ree significant figures4 #c4 ,write out ! fir/ f\r t]ms ( 7a - b7#g4 ,f9d ! f\r? t]m ( ?is expansion :5 a x7-#a7 y7#a_/b77a_/c71 b #ixy7-#d77a_/f71 express+ ! result 9 t]ms ( a s+le radical1 & )\t fractional or negative expon5ts4 #d4 ,r$uce ! foll[+ expression to a polynomial 9 a & b3 7#f;a#c #g;ab#b #ab;b#c7_/7c;a#b - #e;ab - #d;b#b7 - #a_/c_/ai;b - 7#e;a #d;b7_/7#ai;a#b74 #e4 ,! co/ ( publi%+ a book consi/s ( two ma9 items3 fir/1 ! fix$ exp5se ( sett+ up ! type2 &1 second1 ! runn+ exp5ses ( presswork1 b9d+1 etc41 :i* may be assum$ to be proportional to ! numb] ( copies4 ,a c]ta9 book co/s #ce c5ts a copy if #ajjj copies are publi%$ at "o "t1 b only #ai c5ts a copy if #ejjj copies are publi%$ at "o "t4 ,f9d 7a7 ! co/ ( sett+ up ! type = ! book1 & 7b7 ! co/ ( presswork1 b9d+1 etc41 p] ?\s& copies4 ,,h>v>d ,,univ]sity ,,elem5t>y ,,algebra ,,"t3 ,,"o ,,h\r ,,& ,a ,,half #a4 ,f9d ! highe/ common factor & ! l[e/ common multiple ( ! ?ree expressions a#d - b#d2 a#c b#c2 a#c #b;a#b b #b;ab#b b#c4 #b4 ,solve ! quadratic equation x#b - #a4fx #j4c #j1 comput+ ! value ( ! l>g] root correct to ?ree significant figures4 #c4 ,9 ! expression x#b - #bxy y#b - #db7a_/b77x y7 #h1 sub/itute = x & y ! values x 7u v #a7_/b7a_/b71 y 7u - v #a7_/b7a_/b71 & r$uce ! result+ expression to its simple/ =m4 #d4 ,/ate & prove ! =mula = ! sum ( ! fir/ n t]ms ( a geometric progression 9 :i* a is ! fir/ t]m & r ! con/ant ratio4 #e4 ,a /ate legislature is to elect a ,unit$ ,/ates s5ator1 a majority ( all ! votes ca/ be+ necess>y = a *oice4 ,!re are ?ree c&idates1 ,a1 ,b1 & ,c1 & #ajj memb]s vote4 ,on ! fir/ ballot ,a has ! l>ge/ numb] ( votes1 receiv+ #i m votes ?an 8 ne>e/ competitor1 ,b2 b he fails ( ! necess>y majority4 ,on ! second ballot ,c's "n is )drawn1 & all ! memb]s :o vot$ = ,c n[ vote = ,b1 :]eupon ,b is elect$ by a majority ( #b4 ,h[ _m votes 7 ca/ = ea* c&idate on ! fir/ ballot8 ,,massa*usetts ,,in/itute ,,( ,,te*nology ,,algebra ,a ,,"t3 ,,"o ,,h\r ,,& ,,?ree ,,qu>t]s #a4 ,factor ! expressions3 x#c x#b #bx4 x#c x#b - #dx - #d4 #b4 ,simplify ! expression3 #a - 7b#b7_/7;a#b7a - 7ab - b#b7_/7;a#b77;a#d7_/7;a#c b#c7 4 7a - b7_/7a#b b#b74 #c4 ,f9d ! value ( x #a x#b7a_/b71 :5 x 7#a_/b7;a_/b7#a_/b7 - b_/a7#a_/b74 #d4 ,solve ! equations3 7#gx #f7_/aa y - #af 7#ex - #ac7_/b - 7#hy - x7_/#e1 #c7cx #d7 #ajy - #ae4 #e4 ,solve ! equations3 ,a ,c #b1 -,a ,b ,c ,d #a1 #b;,a - ,b #b;,c ,d #e1 ,b ,d #a4 #f4 ,two squ>es are =m$ ) a comb9$ p]imet] ( #af in*es4 ,"o squ>e conta9s #d squ>e in*es m ?an ! o!r4 ,f9d ! area ( ea*4 #g4 ,a man walk$ to a railway /ation at ! rate ( #d miles an h\r & travel$ by tra9 at ! rate ( #cj miles an h\r1 rea*+ 8 de/9ation 9 #bj h\rs4 ,if he _h walk$ #c miles an h\r & ridd5 #ce miles an h\r1 he w\ld h made ! j\rney 9 #ah h\rs4 ,requir$ ! total di/ance travel$4 ,,massa*usetts ,,in/itute ,,( ,,te*nology ,,algebra ,b ,,"t3 ,,"o ,,h\r ,,& ,,?ree ,,qu>t]s #a4 ,h[ _m t]ms mu/ be tak5 9 ! s]ies #b1 #e1 #h1 #aa1 444 s t ! sum %all be #cde8 #b4 ,prove ! =mula x -b - b#b - #d;ac7#a_/b7_/7b;a7 = solv+ ! quadratic equation ax#b bx c #j4 #c4 ,f9d all values ( a = :i* sqa is a root ( x#b x #bj #b;a1 & *eck y\r results4 #d4 ,solve x#b #cy#b #aj1 x - y #b1 & sket* ! graphs4 #e4 ,! sum ( two numb]s x & y is #e1 & ! sum ( ! two middle t]ms 9 ! expansion ( 7x y7#c is equal to ! sum ( ! fir/ & la/ t]ms4 ,f9d ! numb]s4 #f4 ,solve x#d - #bx#c #cx#b - #bx #a #j4 7,,h9t3 ,divide by x#b & sub/itute x #a_/x z47 #g4 ,9 anticipation ( a holiday a m]*ant makes an outlay ( #ej1 :i* w be a total loss 9 case ( ra91 b :i* w br+ him a cle> pr(it ( #aej above ! outlay if ! day is pleasant4 ,to insure aga9/ loss he takes out an insurance policy aga9/ ra9 = a c]ta9 sum ( money = :i* he has to pay a c]ta9 p]c5tage4 ,he !n f9ds t :e!r ! day be ra9y or pleasant he w make #hj cle>4 ,:at is ! am\nt ( ! policy1 & :at rate did ! company *>ge him8 ,,massa*usetts ,,in/itute ,,( ,,te*nology ,,algebra ,a ,,"t3 ,,two ,,h\rs #a4 ,simplify m #a_/m#b n #a_/n#b mn #a_/mn#b - m #a_/mn #a_/nmn #a_/mn4 #b4 ,f9d ! prime factors ( 7a7 7x - x#b7c 7x#b - #a7c 7#a - x7#c4 7b7 7#bx a - b7#d - 7x - a b7#d4 #c4 7a7 ,simplify 7xq7_/7xr77q r7 7xr7_/7xp77r p7xp_/xq7p q74 7b7 ,%[ t 7x#a_/7n#a77a_/n77_/7x#a_/7n#b7a_/7n#a77 x7#a_/n7 4 x#a_/7n#b7_/x#ba_/7n#a74 #d4 ,def9e homog5e\s t]ms4 ,= :at value ( n is xn y7#e - n_/#b7 x7n #a7 y7#bn - #f7 a homog5e\s b9omial8 #e4 ,extract ! squ>e root ( x7x - #b7a_/b777x - #h7a_/b777x - #ah7a_/b77 #d4 #f4 ,two vessels conta9 ea* a mixture ( w9e & wat]4 ,9 ! fir/ vessel ! quantity ( w9e is to ! quantity ( wat] z #a 3 #c1 & 9 ! second z #c 3 #e4 ,:at quantity mu/ be tak5 f ea*1 s z to =m a ?ird mixture :i* %all conta9 #e gallons ( w9e & #i gallons ( wat]8 #g4 ,f9d a quantity su* t by add+ x to ea* ( ! quantities a1 b1 c1 d1 we obta9 f\r quantities 9 proportion4 #h4 ,:at values mu/ be giv5 to a & b1 s t 7#c;a #b;b #ag7_/b1 7#b;a - #c;b #be7_/c1 #d - #e;a - #ac;b may be equal8 ,,m\nt ,,holyoke ,,college ,,elem5t>y ,,algebra ,,"t3 ,,two ,,h\rs #a4 ,factor ! foll[+ expressions3 7a7 a7#c_/d7 - b7#c_/d74 7b7 x#b y#b z#b - x#b z - y#b z #a4 7c7 #af7x y7#d - 7#bx - y7#d4 #b4 7a7 ,simplify 7a#b b#b77;b#d7_/7;b#b - a#b7 - a#b_/;a_/7a b7 b_/7a - b74 7b7 ,extract ! squ>e root ( x#d - #bx#c #ex#b - #dx #d4 #c4 ,solve ! foll[+ equations3 7a7 #a_/x #a_/y #e1 #a_/7x#b7 #a_/7y#b7 #ac4 7b7 x#b - #ex #b #j4 7c7 #bgx #a7a_/b7 #b - #ccx7#a_/b74 #d4 ,simplify3 7a7 #ged7a_/c7 #bef7a_/f7 #dcb_/7-bej77a_/c74 7b7 #a_/7;a - b77b - c7 #a_/7;c - a77b - a74 7c7 ,f9d #ai - #hc7a_/b77a_/b74 #e4 ,plot ! graphs ( ! foll[+ sy/em1 & det]m9e ! solution f ! po9t ( int]section3 x - #by #j1 #bx - #cy #d4 #f4 7a7 ,d]ive ! =mula = ! solution ( ax#b bx c #j4 7b7 ,det]m9e ! value ( m = :i* ! roots ( #bx#b #dx m #j are 7i7 equal1 7ii7 real1 7iii7 imag9>y4 7c7 ,=m ! quadratic equation ^: roots are #b #c7a_/b7 & #b - #c7a_/b74 #g4 ,a page is to h a m>g9 ( #a in*1 & is to conta9 #ce squ>e in*es ( pr9t+4 ,h[ l>ge mu/ ! page be1 if ! l5g? is to exce$ ! wid? by #b in*es8 #h4 7a7 ,9 an ari?metical progression ! sum ( ! fir/ six t]ms is #bfa1 & ! sum ( ! fir/ n9e t]ms is #big4 ,f9d ! common diff]5ce4 7b7 ,?ree numb]s ^: sum is #bg are 9 ari?metical progression4 ,if #a is add$ to ! fir/1 #c to ! second1 & #aa to ! ?ird1 ! sums w be 9 geometrical progression4 ,f9d ! numb]s4 7c7 ,d]ive ! =mula = ! sum ( n t]ms ( a geometrical progression4 #i4 7a7 ,exp& & simplify 7#b;a#b - #cx#c7g4 7b7 ,= :at value ( x w ! ratio #g x 3 #ab x be equal to ! ratio #e 3 #f8 ,,univ]sity ,,( ,,p5nsylvania ,,elem5t>y ,,algebra ,,"t3 ,,?ree ,,h\rs #a4 ,simplify3 7a x7_/7a - x7 - 7a - x7_/7a x7 _/ 7#d;ax7_/7a#b - x#b74 #b4 ,f9d ! ,h4 ,c4 ,f4 & ,l4 ,c4 ,m4 ( #aj;ab#b7x#b - #b;ax71 #ae;a#c;b7x#b - ax - #b;a#b71 #be;b#c7x#b - a#b7b4 #c4 ,a groc] buys eggs at #d = #g;c4 ,he sells #a_/d ( !m at #e = #ab;c1 & ! re/ at #f = #aa;c1 mak+ #bg;c by ! transaction4 ,h[ _m eggs does he buy8 #d4 ,solve = t3 7t #d;a b7_/7t a b7 - 7#dt - a - #b;b7_/7t a - b7 -#c4 #e4 ,f9d ! squ>e root ( a#b - 7#c_/b7;a7#c_/b7 - 7#c_/b7;a7#a_/b7 7#da_/af7;a #a4 #f4 7a7 ,= :at values ( m w ! roots ( #bx#b #cmx -#b be equal8 7b7 ,if #b;a #c;b is a root ( x#b - #f;bx - #d;a#b #i;b#b #j1 f9d ! o!r root )\t solv+ ! equation4 #g4 7a7 ,solve = x3 #bx - #c;a7#a_/b7 #cx - #b;a7#a_/b7 #c;a7#a_/b74 7b7 ,solve = m3 #a - 7#a7_/7b - m7 #a_/7m #b7 7m - #f7_/7d - m#b74 #h4 ,solve ! sy/em3 x#b #by#b #ag2 xy - y#b #b4 #i4 ,two boats leave simultane\sly opposite %ores ( a riv] #b-a_/d mi4 wide & pass ea* o!r 9 #ae m94 ,! fa/] boat completes ! trip #f-c_/d m94 be=e ! o!r rea*es ! opposite %ore4 ,f9d ! rates ( ! boats 9 miles p] h\r4 #aj4 ,write ! six? t]m ( x_/7#by#b7a_/c77 - 7y7#a_/b77_/x#i )\t writ+ ! prec$+ t]ms4 #aa4 ,! sum ( ! #b;d & #bj? t]ms ( an ,a4 ,p4 is #aj1 & _! product is #bc-dg_/fd4 ,:at is ! sum ( sixte5 t]ms8 ,,pr9ceton ,,univ]sity ,,algebra ,a ,,"t3 ,,two ,,h\rs ,c&idates :o are at ?is "t tak+ bo? ,algebra ,a & ,algebra ,b may omit f ,algebra ,a que/ions #d1 #e1 & #f1 & f ,algebra ,b que/ions #a 7a71 #c1 & #d4 #a4 ,simplify 7a#c a#b;b ab#b7_/7;a#b - #c;ab - #d;b#b7 _/ 7a#b #f;ab - #g;b#b7_/7;a#b #h;ab - #i;b#b7 4 7a#c - b#c7_/7;a#b - #g;ab #ab;b#b74 #b4 7a7 ,divide a7#e_/b7 ab7#c_/b7 b7#e_/b7 - #b;a7#a_/b7;b#b - a7#c_/b7;b by a7#c_/b7 - b7#c_/b7 a7#a_/b7;b - ab7#a_/b74 7b7 ,simplify 7#a7_/7x7-#a7 y7-#a7 4 7x7#a_/d7y7#a_/b77c #a4 #c4 ,factor3 7a7 7x#b #cx7#b - 7#bx - #f7b4 7b7 a#b ac - #d;b#b - #b;bc4 #d4 ,solve #a_/7x #a7 - 7#a7_/7x - #a7 - 7#a7_/7x - #c7 7#a7_/7x - #e7 #j4 #e4 ,solve = x & y3 mx ax my - by1 x - y a b4 #f4 ,! road f ,a to ,b is uphill = #e mi41 level = #d mi41 & !n d[nhill = #f mi4 ,a man walks f ,b to ,a 9 #d hr42 lat] he walks halfway f ,a to ,b & back aga9 to ,a 9 #c hr4 & #ee m942 & lat] he walks f ,a to ,b 9 #c hr4 & #eb m94 ,:at are 8 rates ( walk+ uphill1 d[nhill1 & on ! level1 if ^! d n v>y8 ,,algebra ,b #a4 ,solve 7a7 7x #a7_/7x - #b7 7#bx #a7_/7x #a7 7#cx #c7_/7a - x7 #j4 7b7 #bx #g7a_/b7 #cx - #ah7a_/b7 - #gx #a7a_/b7 #j4 7c7 #f_/7x#b #bx7 #e - #bx - x#b4 #b4 ,solve = x & y1 *eck+ "o solution 9 ea* problem3 7a7 #bx #cy #a1 #f_/x #a_/y #b4 7b7 x#b x y1 y#b #cy - x4 #c4 ,a man arranges to pay a debt ( #cfjj 9 #dj mon?ly paym5ts :i* =m an ,a4 ,p4 ,aft] pay+ #cj ( !m he /ill owes #a_/c ( 8 debt4 ,:at 0 8 fir/ paym5t8 #d4 ,if #d quantities are 9 proportion & ! second is a mean proportional betwe5 ! ?ird & f\r?1 prove t ! ?ird w be a mean prop4 betwe5 ! fir/ & second4 #e4 ,9 ! expansion ( #bx #a_/cx#f ! ratio ( ! f\r? t]m to ! fif? is #b 3 #a4 ,f9d x4 #f4 ,two m5 ,a & ,b c toge!r d a piece ( "w 9 #ab days2 ,b w\ld ne$ #aj days m ?an ,a to d ! :ole "w4 ,h[ _m days w\ld x take ,a alone to d ! "w8 ,,algebra ,,to ,,quadratics #a4 ,simplify 7ab7-#b7;c#b77a_/b7 4 7a#c;b#b;c7-#c777a_/c7 7a#f7_/7;b77#a_/c74 #b4 ,simplify a_/7a - b77a - c7 b_/7b - c77b - a7 c_/7c - a77c - b74 #c4 ,factor 7a7 x#d - #ajx#b #i4 7b7 x#b #bxy - a#b - #b;ay4 7c7 7a b7#b 7a c7#b - 7c d7#b - 7b d7#b4 #d4 ,f9d ,h4 ,c4 ,f4 ( x#d - x#c #bx#b x #c & 7x #b77x#c - #a74 #e4 ,solve x_/7x - #b7 7x - #i7_/7x - #g7 7x #a7_/7x - #a7 7x - #h7_/7x - #f74 #f4 ,! sum ( ?ree numb]s is #ea2 if ! fir/ numb] be divid$ by ! second1 ! quoti5t is #b & ! rema9d] #e2 if ! second numb] be divid$ by ! ?ird1 ! quoti5t is #c & ! rema9d] #b4 ,:at are ! numb]s8 ,,smi? ,,college ,,elem5t>y ,,algebra #a4 ,factor e7#bx7 - #b e7-#bx71 x7#ab7 - #h1 x#b - x - y#b - y1 #ah;a#bx#b -#bd;axy - #ajy#b4 #b4 ,solve #g #dx #cbx#b #ex #g7a_/b77a_/b7 - #c #j4 #c4 ,! second t]m ( a geometrical progression is #cb7a_/b71 & ! fif? t]m is #c_/af4 ,f9d ! fir/ t]m & ! ratio4 #d4 ,solve ! foll[+ equations & *eck y\r results by plott+3 x#b y#b - xy #g1 x y #d4 #e4 ,solve #a_/7x#c7 #a_/7y#c7 #bdc_/h1 #a_/x #a_/y #i_/b4 #f4 ,9 an ari?metical progression d -#aa1 n #ac1 s #j4 ,f9d a & l4 #g4 ,exp& by ! b9omial !orem & simplify3 7#bx7_/7y#c7 - 7y#d7_/7x#e -#f7a_/b77e4 #h4 ,! diagonal ( a rectangle is #ac ft4 long4 ,if ea* side 7 long] by #b ft41 ! area w\ld be increas$ by #ch sq4 ft4 ,f9d ! l5g?s ( ! sides4 ,,smi? ,,college ,,elem5t>y ,,algebra #a4 ,f9d ! ,h4 ,c4 ,f4 ( #hx#c - #bg1 #cbx#e - #bdc1 & #fx#c - #ix#b #dx - #f4 #b4 ,solve3 7a7 7#bx #e77-e7 #ca7bx #e77-e_/b7 #cb4 7b7 7x - #a77a_/b7 7#cx #a77a_/b7 #d4 #c4 ,a f>m] sold a horse at #ge = :i* he _h paid x doll>s4 ,he realiz$ x p] c5t pr(it by 8 sale4 ,f9d x4 #d4 ,f9d ! #ac? t]m & ! sum ( #ac t]ms ( ! ari?metical progression 7#b7a_/b7 - #a7_/b1 7#b7a_/b77_/b1 7#a7_/b7b7a_/b7 - #a71 4444 #e4 ,! diff]5ce betwe5 two numb]s is #dh4 ,_! ari?metical mean exce$s _! geometrical mean by #ah4 ,f9d ! numb]s4 #f4 ,exp& by ! b9omial !orem & simplify #c;a7-#b7 - a_/-#b7a_/b7e4 #g4 ,solve3 #a_/x #a_/y #c_/b1 #a_/7x#b7 #a_/7y#b7 #e_/d4 #h4 ,solve ! foll[+ equations & *eck ! results by f9d+ ! int]sections ( ! graphs ( ! two equations3 x#b #dy1 x #by #d4 ,,vass> ,,college ,,elem5t>y ,,& ,,int]m$iate ,,algebra ,answ] any six que/ions4 #a4 ,f9d ! product ( #a #b;a_/#c - 7#e;a#b7_/7f7 & #b - #c;a_/#d 7a#b7_/7c74 #b4 ,resolve into l9e> factors3 7a7 #dx#b - #be2 7b7 #fx#b - x - #ab2 7c7 a#b;b#b #a - a#b - b#b2 7d7 y#c 7x - #c7y#b - 7#cx - #b7y #bx4 #c4 ,r$uce to simple/ =m3 7a7 z_/7#a_/x - #a_/y7 y_/7#a - y_/x7 - x_/7#a - x_/y74 7b7 -7x#c77a_/b77a_/c7 x 7#dy7-#c777a_/b74 #d4 7a7 ,divide x7#c_/b7 - x7-#c_/b7 by x7#a_/b7 - x7-#a_/b74 7b7 ,f9d correct to "o place ( decimals ! value ( #e7a_/b7 #g7a_/b7_/b - #c7a_/b74 #e4 7a7 ,if a_/b c_/d1 %[ t 7a#b c#b7_/7;b#b d#b7 ac_/bd4 7b7 ,two numb]s are 9 ! ratio #c 3 #d1 & if #g be subtract$ f ea* ! rema9d]s are 9 ! ratio #b 3 #c4 ,f9d ! numb]s4 #f4 ,solve ! equations3 7a7 7x #a7_/7b7 - #c_/x x_/#c - 7#e - x7_/7#f74 7b7 #aax#b - #aa-a_/d #ix4 7c7 x#b - #by#b #ga1 x y #bj4 #g4 ,a field c\ld be made into a squ>e by dim9i%+ ! l5g? by #aj feet & increas+ ! bread? by #e feet1 b its area w\ld !n be dim9i%$ by #baj squ>e feet4 ,f9d ! l5g? & ! bread? ( ! field4 ,,vass> ,,college ,,elem5t>y ,,& ,,int]m$iate ,,algebra ,answ] six que/ions1 includ+ ,no4 #e & ,no4 #g or #h4 ,c&idates 9 ,int]m$iate ,algebra w answ] ,nos4 #e-i4 #a4 ,f9d two numb]s ^: ratio is #c & su* t two sev5?s ( ! l>g] is #ae m ?an "o half ! small]4 #b4 ,det]m9e ! factors ( ! l[e/ common multiple ( #cx#d 7x#c - y#c71 #ae 7x#d - #bx#by#b y#d71 & #ajy 7x#d x#by#b y#d74 #c4 ,f9d to two decimal places ! value ( #d;a7-#b_/e7 b#j;ab7-#a77a_/b77a_/b71 :5 a -#cb & b -#h4 #d4 ,solve ! equations3 #bx #ey #he1 #by #ez #ajc1 #bz #ex #eg4 #e4 ,solve any #c ( ^! equations3 7a7 x#b #dd - #aex #j4 7b7 #b_/x - x_/#e x_/#bj - #bbc_/cj4 7c7 x#b #hx - #dx#b #cbx #ab7a_/b7 #ba4 7d7 #e_/7x #a7 #h_/7x - #b7 #ab_/7dj - #bx74 #f4 ,! sum ( two numb]s is #ac1 & ! sum ( _! cubes is #iaj4 ,f9d ! small] numb]1 correct to ! second decimal place4 #g4 ,! sum ( #i t]ms ( an ari?metical progression is #df2 ! sum ( ! fir/ #e t]ms is #be4 ,f9d ! common diff]5ce4 #h4 ,expla9 ! t]ms1 & prove t if f\r numb]s are 9 proportion1 !y are 9 proportion by alt]nation1 by inv]sion1 & by composition4 ,f9d x :5 7#c x7_/7#c - x7 7#dj x#c7_/7dj - x#c74 #i4 ,f9d ! value ( x 9 ea* ( ^! equations3 7a7 #gx7#a_/d7 - #cx7#a_/b7 #b4 7b7 7x#b #b77e_/b7 #c_/x#b #b7a_/b7 #dx#b #h4 ,,yale ,,univ]sity ,,algebra ,a ,,"t3 ,,"o ,,h\r ,omit "o "q 9 ,gr\p ,,ii & "o 9 ,gr\p ,,iii4 ,cr$it w be giv5 = six que/ions only4 ,gr\p ,i #a4 ,resolve into prime factors3 7a7 #fx#b - #gx - #bj2 7b7 7x#b - #ex7#b - #b7x#b - #ex7 - #bd2 7c7 a#d #d;a#b #af4 #b4 ,simplify #e - 7a#b - #aix#b7_/7;a#b - #dx#b7 _/ #c - 7a - #ex7_/7a - #bx74 #c4 ,solve #b7x - #g7_/7x#b #cx - #bh7 7#b - x7_/7#d - x7 - 7x #c7_/7x #g7 #j4 ,gr\p ,,ii #d4 ,simplify #b7a_/b7 #bc7a_/b7_/b7a_/b7 - #ab7a_/b71 & compute ! value ( ! fraction to two decimal places4 #e4 ,solve ! simultane\s equations x7-#a_/b7 #by7-#a_/b7 #g_/f1 #bx7-#a_/b7 - y7-#a_/b7 #b_/c4 ,gr\p ,,iii #f4 ,two numb]s are 9 ! ratio ( c 3 d4 ,if a be add$ to ! fir/ & subtract$ f ! second1 ! results w be 9 ! ratio ( #c 3 #b4 ,f9d ! numb]s4 #g4 ,a deal] has two k9ds ( c(fee1 wor? #cj & #dj c5ts p] p\nd4 ,h[ _m p\nds ( ea* mu/ be tak5 to make a mixture ( #gj p\nds1 wor? #cf c5ts p] p\nd8 #h4 ,a1 ,b1 & ,c c d a piece ( "w 9 #cj h\rs4 ,a c d half z mu* aga9 z ,b1 & ,b two ?irds z mu* aga9 z ,c4 ,h[ long w\ld ea* require to d ! "w alone8 ,,yale ,,univ]sity ,,algebra ,b ,,"t3 ,,"o ,,h\r ,omit "o "q 9 ,gr\p ,i & "o 9 ,gr\p ,,ii4 ,cr$it w be giv5 = five que/ions only4 ,gr\p ,i #a4 ,solve 7x a7_/7x b7 7x b7_/7x a7 #e_/b4 #b4 ,solve ! simultane\s equations x#by#b #bhxy - #dhj #j1 #bx y #aa4 ,arrange ! roots 9 correspond+ pairs4 #c4 ,solve #cx7-#c_/b7 #bjx7-#c_/d7 #cb4 ,gr\p ,,ii #d4 ,9 go+ #gejj yd4 a front :eel ( a wagon makes #ajjj m revolutions ?an a re> "o4 ,if ! :eels 7 ea* #a yd4 great] 9 circumf]5ce1 a front :eel w\ld make #fbe m revolutions ?an a re> "o4 ,f9d ! circumf]5ce ( ea*4 #e4 ,two c>s ( equal spe$ leave ,a & ,b1 #bj mi4 ap>t1 at diff]5t times4 ,j z ! c>s pass ea* o!r an accid5t r$uces ! p[] & _! spe$ is decreas$ #aj mi4 p] h\r4 ,"o c> makes ! j\rney f ,a to ,b 9 #ef m941 & ! o!r f ,b to ,a 9 #gb m94 ,:at is _! common spe$8 ,gr\p ,,iii #f4 ,write 9 ! simple/ =m ! la/ ?ree t]ms ( ! expansion ( 7#d;a7#c_/b7 - a7#a_/b7 x7#a_/c77h4 #g4 7a7 ,d]ive ! =mula = ! sum ( an ,a4 ,p4 7b7 ,f9d ! sum to inf9ity ( ! s]ies #a1 -#a_/b1 #a_/d1 -#a_/h1 4444 ,also f9d ! sum ( ! positive t]ms4