Algebra
The Quadratic Formula: Roots and the Discriminant
Updated October 2, 2026
Published by SolveCalc (Operator of solvecalc.live)
For ax² + bx + c = 0 (with a ≠ 0), the solutions are x = (−b ± √(b² − 4ac)) ÷ (2a). The expression under the square root—the discriminant—tells you how many real roots you get.
What each letter means
- a — coefficient of x²
- b — coefficient of x
- c — constant term
Write the equation in standard form first. If you see x² − 5x = −6, move everything to one side: x² − 5x + 6 = 0.
The discriminant
Let D = b² − 4ac.
| Discriminant | Real roots |
|---|---|
| D > 0 | Two distinct real roots |
| D = 0 | One real root (a repeated root) |
| D < 0 | No real roots (complex roots exist, beyond this article) |
Worked example
Solve x² − 5x + 6 = 0.
- Identify a = 1, b = −5, c = 6.
- Discriminant: (−5)² − 4(1)(6) = 25 − 24 = 1.
- Roots: x = (5 ± √1) ÷ 2.
- So x = (5 + 1) ÷ 2 = 3, and x = (5 − 1) ÷ 2 = 2.
Check: (x − 2)(x − 3) = x² − 5x + 6. Or use the Quadratic Solver for the same coefficients.
Another check: factoring vs formula
When D is a perfect square, factoring is often faster. The formula still works and is the reliable path when factoring is messy.
Related algebra on CalcSolver
Evaluating polynomials at a point is a different skill—see The Remainder Theorem Made Simple. For general powers and roots while you work, use the scientific calculator.