Enter the coefficients of ax² + bx + c = 0.
Enter valid numbers for a, b, and c.
The quadratic formula, as taught at GCSE and A-level
For an equation in the form ax² + bx + c = 0, with a not equal to 0, working out the discriminant b² - 4ac first tells you what kind of roots to expect before you calculate a single value. This is the order UK maths teaching follows from GCSE Higher tier through to A-level: find the discriminant, read its sign, then apply the quadratic formula. The calculator above follows the same three steps with the coefficients you enter.
| Discriminant | Roots |
|---|---|
| Greater than 0 | Two distinct real roots |
| Equal to 0 | One repeated real root (the graph touches the x-axis once) |
| Less than 0 | Two complex conjugate roots (the graph never crosses the x-axis) |
Where this tool falls short
This solver only handles one quadratic equation with real number coefficients at a time. It does not factorise expressions, complete the square for you step by step, solve simultaneous equations, or handle cubic and higher-degree polynomials. When the discriminant is negative, the roots are given in the form p plus or minus qi, but the tool does not plot the equation or convert the roots to polar form.
The formula in full
x = (-b plus or minus the square root of the discriminant) divided by 2a. When a is 0 the equation stops being quadratic and becomes linear, so the calculator solves bx + c = 0 instead and returns a single root. Pair this with the Scientific Calculator for other working, or practise the underlying arithmetic with the Mental Arithmetic Trainer.