Solving quadratic equations: factorising, the formula and more

What is a quadratic equation?

A quadratic equation is an equation where the highest power of x is x², such as x² − x − 12 = 0. Solving it means finding the values of x that make it true. These are also called the roots, and on a graph of y = x² − x − 12 they're the points where the curve crosses the x axis.

A quadratic equation usually has two solutions. Sometimes both solutions are the same number, and sometimes there are none at all, because the curve never reaches the x axis.

Solving quadratic equations builds on solving equations with x on its own, and on factorising quadratics. The key stage 4 programme of study covers Years 10 and 11 together and doesn't say which year each topic comes in, so the school's scheme of work is the place to check.

How do we solve quadratic equations?

First rearrange the equation so one side is 0, with the x² term first. Then use one of three methods. The GCSE content shows which are for every student and which are higher tier, marked for the more highly attaining students only:

Method	GCSE content
Factorising	every student
Reading a graph (approximate answers)	every student
The quadratic formula	higher tier
Completing the square	higher tier

Factorising is usually quickest when it works. The quadratic formula solves any quadratic equation that has solutions, so it's the one to use when no factor pair fits. Equations that need rearranging before they can be solved are also higher tier.

How to factor and solve quadratic equations

To solve a quadratic equation by factorising, factorise it into two brackets, then set each bracket equal to 0. This works because if two numbers multiply to make 0, one of them must be 0.

Worked example: solve x squared − x − 12 = 0

1. Factorise. Two numbers that multiply to make −12 and add to make −1 are −4 and 3, so (x − 4)(x + 3) = 0.
2. Either x − 4 is 0, so x = 4, or x + 3 is 0, so x = −3.

Check by substituting each answer back in. For x = 4: 16 − 4 − 12 is 0. For x = −3: 9 + 3 − 12 is 0. Both work.

Watch the signs at the end. The bracket (x + 3) gives x = −3, not 3, because −3 + 3 = 0.

Solving factored quadratic equations

If the equation is already factorised, such as (x − 2)(x + 7) = 0, skip straight to the second step: x = 2 or x = −7. A difference of two squares works the same way: x² − 49 = 0 is (x + 7)(x − 7) = 0, so x = 7 or x = −7.

Worked example: solve x squared + 2x = 8

This one needs rearranging first, which is higher tier.

1. Take 8 from both sides so one side is 0: x² + 2x − 8 = 0.
2. Factorise: (x + 4)(x − 2) = 0.
3. So x = −4 or x = 2.

Check: 16 − 8 = 8, and 4 + 4 = 8. Both work.

A common slip is dividing both sides by x. In x² = 5x, dividing by x gives x = 5 and loses the other solution. Rearrange instead: x² − 5x = 0, so x(x − 5) = 0, and x = 0 or x = 5.

The formula to solve quadratic equations

The quadratic formula solves any quadratic equation written as ax² + bx + c = 0, where a isn't 0. It's higher tier, and the DfE lists it as a formula students must know: it isn't given in the exam.

x = −b ± √b² − 4ac over 2a

In words: minus b, plus or minus the square root of (b squared take away 4ac), all divided by 2a. The ± means you work it out twice, once adding the square root and once subtracting it, which gives the two solutions. If the number under the square root is negative, there are no solutions.

Worked example: solve x squared + 4x + 1 = 0, to 2 decimal places

1. Read off a, b and c: a = 1, b = 4, c = 1.
2. Work out the part under the square root first: b² − 4ac is 16 − 4, which is 12.
3. So x = −4 ± √12 over 2.
4. With a calculator, √12 is 3.4641 to 4 decimal places. Adding gives −0.5359 ÷ 2, which is −0.26795. Subtracting gives −7.4641 ÷ 2, which is −3.73205.
5. To 2 decimal places, x = −0.27 or x = −3.73.

Check with the unrounded answers. For x = −0.268: x² is 0.072 and 4x is −1.072, and 0.072 − 1.072 + 1 is 0. For x = −3.732: x² is 13.928 and 4x is −14.928, which also gives 0.

Worked example: solve 2x squared − 3x − 4 = 0, to 2 decimal places

1. a = 2, b = −3 and c = −4. Keep the minus signs.
2. b² is (−3)², which is 9. 4ac is 4 lots of 2 lots of −4, which is −32. So b² − 4ac is 9 + 32, which is 41.
3. −b is 3, and 2a is 4, so x = 3 ± √41 over 4.
4. √41 is 6.4031. Adding gives 9.4031 ÷ 4, which is 2.3508. Subtracting gives −3.4031 ÷ 4, which is −0.8508.
5. To 2 decimal places, x = 2.35 or x = −0.85.

Check: for x = 2.3508, 2x² is 11.052 and 3x is 7.052, and 11.052 − 7.052 − 4 is 0. For x = −0.8508, 2x² is 1.448 and −3x is 2.552, which add to 4, and 4 − 4 = 0.

Only round at the end. Rounding the square root early can change the last decimal place. The rounding guide covers decimal places.

Solving quadratic equations by completing the square

Completing the square means rewriting x² + bx as a squared bracket. It's higher tier. It uses the fact that (x + 3)² expands to x² + 6x + 9, as in expanding brackets. So x² + 6x is the same as (x + 3)² − 9. The number in the bracket is half the coefficient of x, and you take away its square.

Worked example: solve x squared + 6x − 7 = 0

1. Halve 6 to get 3, so x² + 6x is (x + 3)² − 9.
2. The equation becomes (x + 3)² − 9 − 7 = 0, which is (x + 3)² − 16 = 0.
3. Add 16 to both sides: (x + 3)² = 16.
4. Square root both sides, remembering the negative root: x + 3 = 4 or x + 3 = −4.
5. So x = 1 or x = −7.

Check: 1 + 6 − 7 is 0, and 49 − 42 − 7 is 0.

Worked example: solve x squared − 4x − 1 = 0

Halve −4 to get −2: x² − 4x is (x − 2)² − 4. So (x − 2)² − 5 = 0, and (x − 2)² = 5. Then x − 2 = ±√5, so x = 2 + √5 or x = 2 − √5. That's 4.24 or −0.24 to 2 decimal places.

Check: for x = 4.236, x² is 17.944 and 4x is 16.944, and 17.944 − 16.944 − 1 is 0. For x = −0.236, x² is 0.056 and −4x is 0.944, which add to 1, and 1 − 1 = 0.

Completing the square also gives the turning point of the graph. For y = x² + 6x − 7, written as (x + 3)² − 16, the lowest point is at (−3, −16). That's higher tier too.

Solving quadratic equations graphically

To solve a quadratic equation from a graph, draw the curve and read off where it crosses the x axis. The answers are approximate, because you're reading a scale. Using a graph to find approximate solutions is in the GCSE content for every student, and key stage 3 already includes using quadratic graphs to estimate values. A graph of y = x² − x − 12 crosses the x axis at −3 and 4, the two solutions found above.

Common mistakes when solving quadratics

• Not making one side 0 first. x² + 2x = 8 can't be solved by writing x(x + 2) = 8. Rearrange first.
• Wrong sign on the answer. (x + 5) = 0 gives x = −5.
• Dividing by x. It loses the solution x = 0.
• Missing the negative square root. (x + 3)² = 16 means x + 3 is 4 or −4.
• Losing a sign in the formula. When b is negative, −b is positive, and b² is always positive.
• Dividing only part of the top by 2a. Everything above the line is divided by 2a, not only the square root.

How to help at home

• Ask for the check. Substituting each answer back into the question is the best habit there is, and it catches almost every slip.
• Ask "what makes this bracket 0?" It's the step children rush most.
• Write a, b and c down first before using the formula, signs included.
• Use the calculator carefully. Work out the part under the square root first, and round only at the end.

Solving quadratic equations questions to try

1. Solve (x − 3)(x + 7) = 0.
2. Solve x² + 8x + 15 = 0.
3. Solve x² − 2x − 24 = 0.
4. Solve x² − 81 = 0.
5. Higher tier: solve x² = 7x.
6. Higher tier: solve x² + 2x − 4 = 0 using the quadratic formula, to 2 decimal places.

Answers

1. x = 3 or x = −7
2. (x + 3)(x + 5) = 0, so x = −3 or x = −5
3. (x − 6)(x + 4) = 0, so x = 6 or x = −4
4. (x + 9)(x − 9) = 0, so x = 9 or x = −9
5. x² − 7x = 0, so x(x − 7) = 0, and x = 0 or x = 7
6. a = 1, b = 2, c = −4, so b² − 4ac is 4 + 16, which is 20. x = −2 ± √20 over 2, so x = 1.24 or x = −3.24

Quadratic equations also come up when solving equations with algebraic fractions. For the rest of GCSE maths, see the GCSE maths foundation and GCSE maths higher guides.