Quadratic Equations: Factoring, Completing the Square, and the Quadratic Formula

A quadratic equation is any equation that can be written in the form ax² + bx + c = 0 where a ≠ 0. The highest power of the variable is 2, which gives the equation its name (quadratic comes from "quad" meaning square). Quadratic equations have at most two solutions.

The quadratic formula is x = (-b ± √(b²-4ac)) / (2a). It gives the solutions to any quadratic equation ax² + bx + c = 0 and is derived by completing the square on the general form. It always works, even when factoring is difficult or impossible.

The discriminant is b² - 4ac, the expression under the square root in the quadratic formula. If it is positive, there are two distinct real solutions. If zero, there is exactly one real solution (a repeated root). If negative, there are two complex conjugate solutions and no real solutions.

Use factoring when the equation factors easily with integer coefficients. Use the quadratic formula when factoring is not obvious or when the equation has irrational or complex roots. Completing the square is useful when you need to rewrite the equation in vertex form.

Complex roots occur when the discriminant is negative. They always come in conjugate pairs: a + bi and a - bi. They mean the parabola does not cross the x-axis. Complex roots involve the imaginary unit i where i² = -1.

Quadratic equations model projectile motion (height vs time), area optimization problems, profit maximization, signal processing, and any situation where the relationship between variables involves squaring. The parabolic shape appears in satellite dishes, bridges, and headlight reflectors.