Quadratic Equations and Factoring

Factoring

Find two numbers that multiply to $c$ and add to $b$ (when $a = 1$).

$x^2 + 5x + 6 = 0$ → $(x + 2)(x + 3) = 0$ → $x = -2$ or $x = -3$.

Quadratic Formula

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Discriminant

$\Delta = b^2 - 4ac$: positive → 2 real roots, zero → 1 root, negative → no real roots.

Special Patterns

• $a^2 - b^2 = (a+b)(a-b)$ (difference of squares)
• $a^2 + 2ab + b^2 = (a+b)^2$ (perfect square)

$y = a(x - h)^2 + k$. Vertex at $(h, k)$.

Axis of symmetry: $x = -\frac{b}{2a}$.