Get zero on one side, factorise, set each bracket to zero, and decide which solutions make sense in context. It is on Foundation and Higher, aimed at grade 5. Free questions with answers are on this page. Worked examples show each line before you try the questions.
A quadratic equation is usually solved by factorising to (x−p)(x−q)=0, then x=p or x=q. The zero-product rule is the whole point: a product is zero only if one of the factors is. If there is no constant term, factor out x: x2−5x=0 gives x(x−5)=0, so x=0 or x=5. Do not divide through by x and lose the x=0 root.
If it will not factorise, use the formula or complete the square. 'Give your answers to 2 decimal places' is that hint. Sketching y=x2+… crossing the x-axis is the same two roots.
Rearrange to =0, factorise, then each bracket =0. Never divide by x (you lose the x=0 solution). x2=9 gives x=±3. If it will not factorise, the formula is the next topic.
Objectives
Solve a quadratic with two non-zero roots.
Solve a difference of squares.
Solve when one root is zero, and pick the positive root.
Key points
Rearrange a quadratic equation so it equals zero before factorising.
Factorise and use 'if two things multiply to give 0, one of them is 0'.
Solve quadratics with no constant term, such as x2=7x, by factorising out x.
Solve x2−16=0 and similar using the difference of two squares or square rooting.
Worked examples
Example 1
Solve x^2+5x+6=0.
2 marks
Factorise the quadratic: (x+2)(x+3) = 0.
Set each bracket to zero: x + 2 = 0 or x + 3 = 0, giving x = -3 or -2.
The roots are -3 or -2.
Answer: -3 or -2
Common slips
Slip: Write the roots directly as the coefficients 5 and 6. Factorisation finds values that make the product zero.
Slip: Milo wrote (x+2)(x+3) as the answer. The question asks to solve for x, not just factorise.
Slip: Conclude (x+2)(x+3). Setting factors to zero gives x = -3 or x = -2.
Example 2
Solve x^2-9=0.
2 marks
Recognise difference of squares: (x-3)(x+3) = 0, or x^2 = 9.
Take both square roots: x = -3 or 3.
The roots are -3 or 3.
Answer: -3 or 3
Common slips
Slip: Take 9 as the only root. A quadratic equation has two roots.
Slip: Nina wrote 9. 9 is the square, so x must be ±3.
Slip: Conclude 9. Both -3 and +3 satisfy x^2 = 9.
Example 3
Solve x^2-x-6=0 and give the positive root.
3 marks
Factorise the quadratic: (x-3)(x+2) = 0.
Find both roots: x = 3 or x = -2, then select the positive root 3.
The positive root is 3.
Answer: 3
Common slips
Slip: Add 6 to both sides to get x(x - 1) = 6, then use 6 as a root. The zero-product rule only works when one side is 0. Factorise with the right side equal to 0.
Slip: Otto gave -2. The question specifies the positive root only.
Slip: Conclude -2. 3 is greater than zero.
Questions and answers
GCSE Maths solving quadratics by factorising questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
Solve x^2-5x+6=0.
Show answer
Answer: 2 or 3
Factorise: (x-2)(x-3) = 0.
Solve: x = 2 or 3.
M1 — Factorise: (x-2)(x-3) = 0.
A1 — Solve: x = 2 or 3.
Question 2 Easy2 marks
Solve x^2+6x+8=0.
Show answer
Answer: -4 or -2
Factorise: (x+2)(x+4) = 0.
Solve: x = -4 or -2.
M1 — Factorise: (x+2)(x+4) = 0.
A1 — Solve: x = -4 or -2.
Question 3 Medium2 marks
Solve x^2-16=0.
Show answer
Answer: -4 or 4
Rearrange: x^2 = 16.
Take square roots: x = -4 or 4.
M1 — Rearrange: x^2 = 16.
A1 — Take square roots: x = -4 or 4.
Question 4 Medium2 marks
Solve x^2-25=0.
Show answer
Answer: -5 or 5
Rearrange: x^2 = 25.
Take square roots: x = -5 or 5.
M1 — Rearrange: x^2 = 25.
A1 — Take square roots: x = -5 or 5.
Question 5 Hard2 marks
Solve x^2-x-12=0 and give the positive root.
Show answer
Answer: 4
Factorise: (x-4)(x+3) = 0, roots 4 and -3.
Select positive root: 4.
M1 — Factorise: (x-4)(x+3) = 0, roots 4 and -3.
A1 — Select positive root: 4.
Question 6 Hard2 marks
Solve x^2+x-6=0 and give the positive root.
Show answer
Answer: 2
Factorise: (x+3)(x-2) = 0, roots -3 and 2.
Select positive root: 2.
M1 — Factorise: (x+3)(x-2) = 0, roots -3 and 2.
A1 — Select positive root: 2.
Question 7 Exam-style3 marks
Solve x^2+8x=0.
Show answer
Answer: -8 or 0
Factorise common factor x: x(x+8) = 0.
Set each factor to zero: x = 0 or x + 8 = 0.
State roots: x = -8 or 0.
M1 — x(x + 8) = 0
A1 — x = 0
A1 — x = -8
Common mistakes
Most lost marks on this topic are one slip.
Leaving the factorised brackets as the answer
Set each factor equal to zero
Dropping the zero root
x = 0 is a solution when x is a factor
Giving the negative root when the question asks for the positive one
Choose the root above zero
Spot the slip
Solve x^2+5x+6=0. Ada writes the factors and stops. What are the roots?
-3 or -2
(x+2)(x+3)
5 and 6
Show the slip
-3 or -2 Each factor can be zero: x+2=0 gives -2, x+3=0 gives -3.
Exam tips
Be ready to do this without a calculator. Paper 1 is the non-calculator paper.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
Write '=0' clearly before the factorised line — it signals to the marker that you know why the method works.
State the rejected solution and the reason ('length cannot be negative') — that sentence is often a mark.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.