Quadratic Formula Calculator guide
Quadratic Formula Calculator: Solve ax² + bx + c = 0
Use our quadratic formula calculator to solve quadratic equations, with the discriminant, real or complex roots, factored form and vertex.
By the AllCalculator editorial teamUpdated 16 October 20261 min read
A quadratic equation has the form ax² + bx + c = 0. Our Quadratic Formula Calculator solves it and tells you everything about its graph.
How to use the calculator (Quadratic Formula Calculator)
Enter the coefficients a, b and c. For example, x² − 3x − 10 = 0 has a = 1, b = −3 and c = −10.
You'll see the discriminant, both roots, the factored form, the vertex, the line of symmetry and the y-intercept.
The quadratic formula
x = (−b ± √(b² − 4ac)) ÷ 2a
The part under the square root, b² − 4ac, is the discriminant:
| Discriminant | Roots |
|---|---|
| Positive | Two different real roots |
| Zero | One repeated real root |
| Negative | No real roots (two complex roots) |
A worked example
With the calculator's default settings, x² − 3x − 10 = 0:
- Discriminant: (−3)² − 4 × 1 × (−10) = 9 + 40 = 49
- x = (3 ± 7) ÷ 2
- So x = 5 or x = −2
- Factored: (x + 2)(x − 5) = 0
- Vertex: (1.5, −12.25)
Frequently asked questions
What's the vertex?
The turning point of the parabola, at x = −b ÷ 2a. It's a minimum if a is positive and a maximum if a is negative.
Can I solve by factorising instead?
If the roots are whole numbers, factorising is often quicker. The formula works every time.
What are complex roots?
When the discriminant is negative, the roots involve i, the square root of −1. The graph doesn't cross the x-axis.
For other equations, try our Algebra Math Calculator, and to see the graph, use the Graphing Calculator.
Sources and further reading
- GOV.UK: National curriculum in England – mathematicsgov.uk
- Wolfram MathWorld: Quadratic formulamathworld.wolfram.com
Links open official and independent websites in a new tab.
This guide is general information to help you understand your options. It isn't financial, tax, legal or building advice and doesn't take your personal circumstances into account. Figures were checked when the guide was last updated and can change, so confirm anything important with GOV.UK, HMRC, a regulated adviser or a qualified tradesperson.