Quadratic Equations

Grades 9–12 Algebra Quadratics

A quadratic has an x² as its highest power: ax² + bx + c = 0. Its graph is a parabola — the U-curve of thrown balls, satellite dishes, and headlights.

Simplest case: x² = k

x² = 25 → x = ±5 ← two answers, + and −! x² = 10 → x = ±√10 ≈ ±3.16

The zero-product property

If two things multiply to zero, one of them is zero. That's why factoring solves quadratics:

x² + 5x + 6 = 0 (x + 2)(x + 3) = 0 x + 2 = 0 → x = −2 x + 3 = 0 → x = −3 Answers: x = −2 or x = −3

The quadratic formula — the master key

When factoring is impossible, this always works:

−b ± √(b² − 4ac) x = ───────────────── 2a

The part under the root — b² − 4ac — is the discriminant. Positive → 2 answers, zero → 1 answer, negative → no real answer (the parabola never crosses the x-axis).

Vertex The bottom (or top) of the parabola sits at x = −b/(2a). For y = x² − 6x + 5: x = 3 — the axis of symmetry the curve folds over.

Practice

x² = 16 → x = ___ (either root works)

(x + 2)(x + 3) = 0 — one solution is x = ___ (give the larger)

x² + 9 = 0 has:

For y = x² − 6x + 5, the vertex sits at x = ___

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