Quadratic equation solver
Enter the coefficients a, b and c of ax² + bx + c = 0. You get the roots (real or complex), the discriminant, the vertex of the parabola and the working.
Worked example
With the values shown above when the page opens:
x = 2 or x = 3
- Equation: x² − 5x + 6 = 0
- Discriminant D = b² − 4ac = -5² − 4 × 1 × 6 = 1
- D > 0, so there are two real roots: x = (−b ± √D) ÷ 2a
- √D = 1
- x = 2 or 3
How it works
- The quadratic formula gives the roots of ax² + bx + c = 0: x = (−b ± √(b² − 4ac)) ÷ 2a.
- The part under the root, D = b² − 4ac, is the discriminant. If D is positive there are two real roots, if zero one repeated root, and if negative two complex roots.
- The roots add up to −b/a and multiply to c/a, which is a quick way to check your answer.
- The vertex of the parabola is at x = −b ÷ 2a.
Questions
What if a is zero?
Then it is not quadratic. The equation becomes bx + c = 0 and has the single solution x = −c/b.
What are complex roots?
When the discriminant is negative there is no real square root, so the roots involve i, where i² = −1. The graph then never crosses the x-axis.
How can I factor instead of using the formula?
Look for two numbers that multiply to ac and add to b. For x² − 5x + 6, the numbers are −2 and −3, giving (x − 2)(x − 3).
Related
Calculators are for learning and quick checks. Check important figures yourself, and do not rely on them for tax, legal or financial decisions.