The Quadratic Formula: Steps, Discriminant and Examples
Solve quadratics with the quadratic formula, use the discriminant to predict the answers, and avoid the two most common mistakes.
The quadratic formula
- ax² + bx + c = 0
- x = (−b ± √(b² − 4ac)) / 2a
Get everything on one side first so the equation equals 0.
The discriminant
- b² − 4ac > 0: two real solutions
- b² − 4ac = 0: one real solution
- b² − 4ac < 0: no real solutions
Check it first to know what to expect.
Example 1
Solve x² − 5x + 6 = 0
- a = 1, b = −5, c = 6
- Discriminant: 25 − 24 = 1, so x = (5 ± 1) / 2
x = 3 or x = 2
Example 2
Solve 2x² + 3x − 2 = 0
- a = 2, b = 3, c = −2
- Discriminant: 9 + 16 = 25, so x = (−3 ± 5) / 4
x = 1/2 or x = −2
Common mistake
x = −b ± √(b² − 4ac) / 2a
The whole numerator is divided by 2a, not just the square root.
Watch the sign of −b. If b = −5, then −b = 5.
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