Quadratic Formula Calculator
Solve ax² + bx + c = 0 for real or complex roots. Enter the three coefficients below.
Discriminant Quick Reference
| Discriminant (D) | Root Type | Example |
|---|---|---|
| D > 0 | Two distinct real roots | x² − 5x + 6 = 0 → x = 2, 3 |
| D = 0 | One repeated real root | x² − 4x + 4 = 0 → x = 2 |
| D < 0 | Two complex conjugate roots | x² + x + 1 = 0 → x = −0.5 ± 0.866i |
Frequently Asked Questions
What is the quadratic formula?
The quadratic formula solves any equation of the form ax² + bx + c = 0: x = (−b ± √(b² − 4ac)) / (2a). The ± means there are usually two solutions. The expression under the square root, b² − 4ac, is called the discriminant, and it determines the type and number of roots.
What does the discriminant tell you?
The discriminant D = b² − 4ac determines the nature of the roots: if D > 0, there are two distinct real roots; if D = 0, there is exactly one repeated real root; if D < 0, there are two complex (imaginary) roots. The discriminant is the first thing to check when analysing a quadratic equation.
What are complex roots?
When the discriminant is negative, the square root of a negative number produces an imaginary number. The roots take the form x = p ± qi, where p is the real part (−b / 2a) and q is the imaginary part (√|D| / 2a), and i = √(−1). Complex roots always come in conjugate pairs: if p + qi is a root, so is p − qi.
How do I use the quadratic formula step by step?
Given x² − 5x + 6 = 0 (a=1, b=−5, c=6): (1) Calculate D = (−5)² − 4(1)(6) = 25 − 24 = 1. (2) Since D > 0, there are two real roots. (3) x₁ = (5 + √1) / 2 = 3, x₂ = (5 − √1) / 2 = 2. You can verify: (x − 3)(x − 2) = x² − 5x + 6 ✓
Can every quadratic be solved with the quadratic formula?
Yes — the quadratic formula works for any quadratic equation where a ≠ 0. Other methods (factoring, completing the square) are sometimes faster for 'nice' equations, but the quadratic formula always works. If a = 0, the equation is linear (bx + c = 0), not quadratic.