Quadratic Equations

Quadratic Equations

A quadratic equation has the form ax² + bx + c = 0, where a ≠ 0. These equations have two solutions.

Methods of Solving

Method 1: Factoring

If x² + 5x + 6 = 0, find two numbers that multiply to 6 and add to 5.

(x + 2)(x + 3) = 0

x = −2 or x = −3

Method 2: Quadratic Formula

x = (−b ± √(b² − 4ac)) / 2a

For x² + 5x + 6 = 0: a=1, b=5, c=6

x = (−5 ± √(25 − 24)) / 2 = (−5 ± 1) / 2

x = −2 or x = −3

Method 3: Completing the Square

Rearrange the equation to form (x + p)² = q

Discriminant

The discriminant Δ = b² − 4ac determines the number of solutions:

• If Δ > 0: Two distinct real solutions
• If Δ = 0: One repeated real solution
• If Δ < 0: No real solutions (complex solutions)