Quadratic Equations

Three Forms

1. Standard form: ax² + bx + c = 0

2. Factored form: a(x - r)(x - s) = 0 (r, s are roots)

3. Vertex form: a(x - h)² + k (vertex at (h, k))

Factoring — find two numbers that multiply to ac and add to b

Quadratic formula — works always

Completing the square — useful for vertex form

x = (-b ± √(b² - 4ac)) / 2a

Positive → 2 real solutions

Zero → 1 real solution (double root)

Negative → no real solutions

Vertex = (-b/2a, f(-b/2a)) for standard form

Sum of roots = -b/a, Product of roots = c/a

Watch for disguised quadratics: x⁴ - 5x² + 4 = 0