The quadratic formula solves any quadratic equation ax² + bx + c = 0, including ones that don’t factorise:
x = (−b ± √(b² − 4ac)) ÷ 2a
The ± means you do the calculation twice: once adding the square root and once subtracting it. That gives the two solutions.
Step by step
- Rearrange so the equation is ax² + bx + c = 0, with 0 on one side.
- Write down a, b and c, including their signs.
- Work out b² − 4ac first. This number is called the discriminant.
- Substitute into the formula and simplify.
Worked examples
Example 1: a surd answer
Solve x² − 2x − 1 = 0.
a = 1, b = −2, c = −1. Then b² − 4ac = (−2)² − 4 × 1 × (−1) = 4 + 4 = 8.
x = (2 ± √8) ÷ 2. Since √8 = 2√2, this simplifies to x = 1 ± √2, about −0.414 and 2.414.
See Example 1 in the Quadratic Solver →
Example 2: a is not 1
Solve 2x² + 3x − 2 = 0.
a = 2, b = 3, c = −2. Then b² − 4ac = 9 + 16 = 25.
x = (−3 ± √25) ÷ 4 = (−3 ± 5) ÷ 4, so x = ½ or x = −2.
Because 25 is a square number, this equation also factorises: (2x − 1)(x + 2) = 0.
Example 3
Solve 3x² − 5x + 1 = 0.
a = 3, b = −5, c = 1. Then b² − 4ac = 25 − 12 = 13.
x = (5 ± √13) ÷ 6, about 0.232 and 1.434.
The discriminant
The number under the square root, b² − 4ac, tells you how many solutions there are before you finish the calculation:
- Positive: two different solutions.
- Zero: one repeated solution, because ± 0 changes nothing.
- Negative: no real solutions, because a negative number has no real square root.
For x² + x + 2 = 0, b² − 4ac = 1 − 8 = −7, so there are no real solutions. On a graph, the line in the intersecting lines method misses the parabola completely.
Common mistakes
- Losing the sign of b. If b = −2, then −b = 2. Write b in brackets when you substitute: (−2)².
- Squaring a negative b wrongly. (−2)² = 4, not −4.
- Dividing only part of the top by 2a. The whole of −b ± √(b² − 4ac) is divided by 2a.
- Forgetting to rearrange. For x² = 3x + 4, rewrite it as x² − 3x − 4 = 0 before reading off a, b and c.
Which method should I use?
- Factorising is fastest when it works, which is when b² − 4ac is a square number. Try it first on simple equations.
- The quadratic formula always works, and gives exact surd answers when the equation doesn’t factorise.
- Intersecting lines shows the solutions on a graph. It is ideal when a question asks you to solve graphically, or to see how many solutions there are.
The Quadratic Solver shows all three methods for any equation: type it in, then switch between them.