To solve a quadratic equation, rearrange it so one side is zero, then factorise. If two brackets multiply to give zero, one of them must be zero, so each bracket gives a solution: (x − 3)(x + 4) = 0 means x = 3 or x = −4. The solutions are where the graph of the quadratic crosses the x-axis. When a quadratic does not factorise, use the quadratic formula.
Solving quadratics
- Rearrange to ax² + bx + c = 0 first
- If (x − p)(x − q) = 0, then x = p or x = q
- x² + bx = 0: take out x, giving x = 0 or x = −b
- x² − a² = 0: x = a or x = −a
- Quadratic formula: x = (−b ± √(b² − 4ac)) / 2a
Solving by factorising
Factorise the quadratic into two brackets, then use the fact that if two numbers multiply to give 0, at least one of them is 0. For x² + x − 12 = 0, factorising gives (x + 4)(x − 3) = 0, so x + 4 = 0 or x − 3 = 0. The solutions are x = −4 and x = 3. Watch the signs: the bracket (x + 4) gives x = −4.
Quadratics and their graphs
The graph of a quadratic is a U-shaped curve called a parabola. The solutions of the equation are the x-coordinates where the curve crosses the x-axis, because y = 0 there. A quadratic can have two solutions, one repeated solution, or none, depending on how many times the curve meets the axis.
Special quadratics
If there is no constant term, take out x as a common factor: x² + 7x = 0 becomes x(x + 7) = 0, so x = 0 or x = −7. Do not divide by x, because that loses the solution x = 0. A difference of two squares, such as x² − 36 = 0, gives x = 6 or x = −6.
Quadratics with a coefficient of x² (Higher)
For an equation such as 2x² − 5x − 3 = 0, factorise as (2x + 1)(x − 3) = 0. The bracket (2x + 1) gives 2x = −1, so x = −½. The other gives x = 3.
The quadratic formula (Higher)
If a quadratic does not factorise, use x = (−b ± √(b² − 4ac)) / 2a. For x² + 4x − 1 = 0, a = 1, b = 4 and c = −1, so x = (−4 ± √20)/2 = −2 ± √5. Give answers in surd form or to the accuracy asked.
Worked examples
Example 1. Solve x² + x − 12 = 0.
- Two numbers multiplying to −12 and adding to 1: 4 and −3.
- (x + 4)(x − 3) = 0.
Answer: x = −4 or x = 3.
Example 2. Solve x² − 8x + 15 = 0.
- Multiply to 15 and add to −8: −3 and −5.
- (x − 3)(x − 5) = 0.
Answer: x = 3 or x = 5.
Example 3. Solve x² + 7x = 0.
- Take out x: x(x + 7) = 0.
Answer: x = 0 or x = −7.
Example 4. Solve x² − 36 = 0.
- Difference of two squares: (x − 6)(x + 6) = 0.
Answer: x = 6 or x = −6.
Example 5. Solve 2x² − 5x − 3 = 0.
- Factorise: (2x + 1)(x − 3) = 0.
- 2x + 1 = 0 gives x = −½.
Answer: x = −½ or x = 3.
Example 6. Solve x² + 4x − 1 = 0, giving your answers in surd form.
- a = 1, b = 4, c = −1: x = (−4 ± √(16 + 4))/2.
- √20 = 2√5, so x = (−4 ± 2√5)/2.
Answer: x = −2 ± √5.
Common mistakes
- Not rearranging to zero. x² + x = 12 must become x² + x − 12 = 0 before factorising.
- Sign errors from the brackets. (x + 4) = 0 gives x = −4.
- Dividing by x. For x² + 7x = 0, factorise instead, or you lose x = 0.
- Giving only one root of x² = 36. x can be 6 or −6.
- Errors in the formula. Use brackets for negative values of b and c.
How quadratic equations are examined
Solving quadratics by factorising appears at both tiers of the Edexcel GCSE (1MA1) and the 4MA1 International GCSE; the formula and harder factorising are Higher. Questions ask you to:
- factorise a quadratic, then use it to solve the equation
- read solutions from the graph of a quadratic
- solve equations with no constant term or of the form x² − a² = 0
- use the quadratic formula, often to 2 decimal places (Higher)
Practise with the quadratics worksheet, which has answers, and the algebra revision engine.
You need factorising first. Next, see completing the square and quadratic inequalities.
Frequently asked questions
What are the steps for solving a quadratic equation?
Rearrange it to equal zero, factorise, then set each bracket equal to zero.
Why does each bracket give a solution?
If two numbers multiply to give zero, at least one of them must be zero.
When should I use the quadratic formula?
When the quadratic does not factorise, or when the question asks for answers to a number of decimal places.