How to solve a quadratic equation
- Rearrange your equation into the form ax² + bx + c = 0, so everything is on one side.
- Enter the coefficients a, b and c, including any minus signs. If there's no x term, b is 0, and if there's no constant, c is 0.
- Read the roots, their exact form with square roots (such as (−3 ± √17) ÷ 4) when the coefficients are whole numbers, then the discriminant, vertex, axis of symmetry and whether the curve opens upwards or downwards. The steps show the formula worked through.
Worked example: 2x² − 4x − 6 = 0
Here a = 2, b = −4 and c = −6.
- Discriminant: b² − 4ac = (−4)² − 4 × 2 × (−6) = 16 + 48 = 64
- Square root of the discriminant: √64 = 8
- Roots: x = (4 ± 8) / 4, so x = 12 / 4 = 3 and x = −4 / 4 = −1
Check by putting x = 3 back in: 2 × 9 − 4 × 3 − 6 = 18 − 12 − 6 = 0. The equation also factors as 2(x − 3)(x + 1) = 0, which gives the same roots.
The vertex, the lowest point of this curve, is at x = −b / 2a = 4 / 4 = 1, and y = 2 − 4 − 6 = −8, so it’s (1, −8). The axis of symmetry is the line x = 1, halfway between the two roots. Because a is positive, the parabola opens upwards.
When to use which method
| Method | Best for |
|---|---|
| Factorising | Small whole-number roots you can spot, like x² − 5x + 6 = (x − 2)(x − 3) |
| Square root | No x term: x² − 49 = 0 gives x = ±7 |
| Completing the square | Finding the vertex form a(x − h)² + k by hand |
| Quadratic formula | Everything else, including roots that aren’t whole numbers |
The formula always works, which is why this solver uses it. Roots that aren’t whole are shown as decimals to up to six places. If you need an exact surd like 1 + √3, use the discriminant shown: the roots are −b ± √(discriminant), all over 2a.