How to Solve Quadratic Equations: 3 Methods with Examples
Quick answer: to solve a quadratic equation ax² + bx + c = 0, first try factoring. If the roots are not whole numbers, use the quadratic formula x = (−b ± √(b² − 4ac)) / 2a, which works for every quadratic equation.
What is a quadratic equation?
A quadratic equation is any equation that can be written as ax² + bx + c = 0, where a, b and c are numbers and a ≠ 0. The highest power of x is 2, so there are at most two solutions, also called roots.
Method 1: Factoring
Factoring works when you can find two numbers that multiply to a·c and add to b.
Example: x² − 5x + 6 = 0
- Find two numbers that multiply to 6 and add to −5: −2 and −3.
- Write the factors: (x − 2)(x − 3) = 0.
- Set each factor to zero: x = 2 or x = 3.
Method 2: Completing the square
Completing the square rewrites the equation as (x + p)² = q. It is useful for finding the vertex of a parabola.
Example: x² + 6x − 7 = 0
- Move the constant: x² + 6x = 7.
- Add (6/2)² = 9 to both sides: x² + 6x + 9 = 16.
- Write as a square: (x + 3)² = 16.
- Take square roots: x + 3 = ±4, so x = 1 or x = −7.
Method 3: The quadratic formula
The quadratic formula always works:
x = (−b ± √(b² − 4ac)) / 2a
Example: 2x² − 5x − 3 = 0, so a = 2, b = −5, c = −3.
- Discriminant: b² − 4ac = 25 + 24 = 49.
- x = (5 ± 7) / 4.
- x = 3 or x = −1/2.
Which method should you use?
| Situation | Best method |
|---|---|
| Small whole-number coefficients | Factoring |
| You need the vertex or vertex form | Completing the square |
| Decimals, large numbers or no obvious factors | Quadratic formula |
| Negative discriminant (complex roots) | Quadratic formula |
Use the discriminant to predict the answer
- b² − 4ac > 0: two real roots
- b² − 4ac = 0: one repeated root
- b² − 4ac < 0: two complex roots
Check your answer
Substitute each root back into the original equation. For x = 3 in 2x² − 5x − 3: 18 − 15 − 3 = 0 ✓. You can also run any quadratic through our free quadratic equation solver to see every step.