A quadratic equation has the form ax² + bx + c = 0 where a ≠ 0. It can have two real solutions, one repeated solution, or no real solutions. There are three standard ways to solve it, and the best choice depends on the numbers in front of you.
Method 1: Factoring
Factoring works when you can spot two numbers that multiply to a·c and add to b. Take x² − 5x + 6 = 0. We need two numbers that multiply to 6 and add to −5: those are −2 and −3.
- Rewrite:
(x − 2)(x − 3) = 0 - Zero-product rule: if a product is zero, one factor is zero.
- So
x − 2 = 0orx − 3 = 0, giving x = 2 or x = 3.
Check: 2² − 5·2 + 6 = 4 − 10 + 6 = 0 ✓ and 3² − 5·3 + 6 = 9 − 15 + 6 = 0 ✓.
Method 2: Completing the square
This method always works and is how the quadratic formula is derived. Solve x² + 6x + 5 = 0.
- Move the constant:
x² + 6x = −5 - Add (b/2)² = (6/2)² = 9 to both sides:
x² + 6x + 9 = 4 - Write the left side as a square:
(x + 3)² = 4 - Take square roots:
x + 3 = ±2, so x = −1 or x = −5.
Method 3: The quadratic formula
For any quadratic, x = (−b ± √(b² − 4ac)) / (2a). Solve 2x² + 3x − 2 = 0, where a = 2, b = 3, c = −2.
- Discriminant:
b² − 4ac = 9 − 4·2·(−2) = 9 + 16 = 25 x = (−3 ± 5) / 4- So x = 1/2 or x = −2.
Check x = 1/2: 2·(1/4) + 3/2 − 2 = 0.5 + 1.5 − 2 = 0 ✓.
What the discriminant tells you
b² − 4ac > 0: two different real solutions.b² − 4ac = 0: one repeated real solution.b² − 4ac < 0: no real solutions (two complex ones).
Common mistakes
- Forgetting the ± and reporting only one root.
- Dropping the sign of c when it is negative inside b² − 4ac.
- Dividing only part of the numerator by 2a.