How to use the quadratic formula calculator
Write your equation in the form ax² + bx + c = 0, then enter a, b and c. If a term is missing, enter 0 for it; for x² − 9 = 0, use a = 1, b = 0, c = −9. You get the roots in decimal form and, for whole-number coefficients, in exact form, plus the discriminant, vertex, factored form and a graph.
The quadratic formula
x = (−b ± √(b² − 4ac)) ÷ 2a
The part under the square root, D = b² − 4ac, is the discriminant. It tells you what kind of roots to expect:
- D > 0: two different real roots. The graph crosses the x-axis twice.
- D = 0: one repeated real root. The graph just touches the x-axis.
- D < 0: two complex roots. The graph doesn't reach the x-axis.
Worked example
Solve 2x² − 4x − 6 = 0, so a = 2, b = −4, c = −6.
D = (−4)² − 4 × 2 × (−6) = 16 + 48 = 64, and √64 = 8. x = (4 ± 8) ÷ 4, so x = 3 or x = −1. Factored: 2(x − 3)(x + 1).
The vertex
Vertex = (−b ÷ 2a, c − b² ÷ 4a)
The vertex is the lowest point of the curve when a is positive, or the highest when a is negative. For 2x² − 4x − 6 it's at (1, −8).
Working with fractions or ratios too? Try the percentage calculator.
Frequently asked questions
What if a is 0?
Then the equation isn't quadratic. bx + c = 0 is linear, with the single solution x = −c ÷ b.
Why are some answers given with i?
When the discriminant is negative, the square root of a negative number appears. Mathematicians write √−1 as i, so the roots are complex numbers such as −1 ± 2i.
How accurate are the decimals?
The discriminant is worked out exactly, so the calculator never mistakes two close roots for one. Roots use a method that stays accurate even when one is very small and the other very large, then are rounded to up to 6 decimal places; very small or very large values are shown in scientific form, such as 2.5 × 10⁻⁸. The exact form has no rounding.
Formulas checked 29 October 2026. How we test our calculators. Check important results, and show your own working where your teacher or exam asks for it.