Solving quadratics by factorising

Visual model

Factorise, then set each bracket to zero

x^{2} + 5x + 6

(x + 2)(x + 3)

then set each bracket equal to zero

Gold-standard guide

26 mins

What you will learn

Turn a quadratic into brackets and find both solutions.

Use a clear step-by-step method for solving quadratics by factorising.

Check your answer and avoid the most common exam mistake.

Useful before you start

Core number skillsEarlier algebra skillsShowing clear working

Core knowledge

Know the rule, then use it

These are the short notes. Read each one, then check you can use it in the worked example below.

Method

Look for two numbers that MULTIPLY to c and ADD to b in x² + bx + c

Step 1

Make sure one side equals zero

The equation is already x² + 5x + 6 = 0

Step 2

Find a pair of numbers that multiply to 6 and add to 5

Pairs: (1, 6) sum 7 — no

Step 3

Write the factorised form

(x + 2)(x + 3) = 0

Watch out

Students find a pair that multiplies correctly but add incorrectly (for example 1 × 6 = 6 but 1 + 6 = 7 ≠ 5)

Zero product rule

If AB = 0, then A = 0 or B = 0.

Factorised quadratic

$(x + a)(x + b) = 0 gives \times = -a or \times = -b.$

Worked example

Solve x² + 5x + 6 = 0 by factorising.

Make sure one side equals zero: The equation is already x² + 5x + 6 = 0. If it were not, rearrange first.

Find a pair of numbers that multiply to 6 and add to 5: Pairs: (1, 6) sum 7 — no. (2, 3) sum 5 — yes. Use 2 and 3.

Write the factorised form: (x + 2)(x + 3) = 0

Apply the zero-product property: x + 2 = 0 → x = −2, or x + 3 = 0 → x = −3. Both solutions must be stated.

Final answer

x = −2 or x = −3

Question ladder

Build up to the hardest questions

Do them in order. If you miss a step, read the solution, then redo the question without looking.

Workedreasoning

Solve x² + 5x + 6 = 0 by factorising.

4 marks4 minssolving-quadratics-by-factorising-worked

Show solution

Worked solution

1.Make sure one side equals zero: The equation is already x² + 5x + 6 = 0. If it were not, rearrange first.
2.Find a pair of numbers that multiply to 6 and add to 5: Pairs: (1, 6) sum 7 — no. (2, 3) sum 5 — yes. Use 2 and 3.
3.Write the factorised form: (x + 2)(x + 3) = 0
4.Apply the zero-product property: x + 2 = 0 → x = −2, or x + 3 = 0 → x = −3. Both solutions must be stated.

Final answer

x = −2 or x = −3

Mark points

M1: make sure one side equals zero
M1: find a pair of numbers that multiply to 6 and add to 5
M1: write the factorised form
M1: apply the zero-product property
A1: x = −2 or x = −3

Watch out

g. 1 × 6 = 6 but 1 + 6 = 7 ≠ 5). Always write the pair check explicitly.Also, ensure the equation equals zero — rearranging first is essential if it does not.

Diagnosticrecall

Solve x² + 7x + 12 = 0

1 mark2 minssolving-quadratics-by-factorising-q1

Show solution

Worked solution

1.Spot the skill: Look for two numbers that MULTIPLY to c and ADD to b in x² + bx + c.
2.Use the make sure one side equals zero stage first, then find a pair of numbers that multiply to 6 and add to 5.
3.Keep the final answer visible: x = −3 or x = −4.

Final answer

x = −3 or x = −4

Mark points

M1: use the correct look for two numbers that multiply to c and add to b in x² + bx + c.write as (x + p)(x + q) = 0, then set each bracket to zero.for 2ax² + bx + c: multiply a by c, find two numbers multiplying to ac and adding to b, split the middle term.
A1: x = −3 or x = −4

Watch out

g. 1 × 6 = 6 but 1 + 6 = 7 ≠ 5). Always write the pair check explicitly.Also, ensure the equation equals zero — rearranging first is essential if it does not.

Easyprocedure

Solve x² − x − 6 = 0

2 marks3 minssolving-quadratics-by-factorising-q2

Show solution

Worked solution

1.Spot the skill: Look for two numbers that MULTIPLY to c and ADD to b in x² + bx + c.
2.Use the find a pair of numbers that multiply to 6 and add to 5 stage first, then write the factorised form.
3.Keep the final answer visible: x = 3 or x = −2.

Final answer

x = 3 or x = −2

Mark points

M1: use the correct look for two numbers that multiply to c and add to b in x² + bx + c.write as (x + p)(x + q) = 0, then set each bracket to zero.for 2ax² + bx + c: multiply a by c, find two numbers multiplying to ac and adding to b, split the middle term.
A1: x = 3 or x = −2

Watch out

g. 1 × 6 = 6 but 1 + 6 = 7 ≠ 5). Always write the pair check explicitly.Also, ensure the equation equals zero — rearranging first is essential if it does not.

Mediumreasoning

Solve x² − 9x + 18 = 0

3 marks4 minssolving-quadratics-by-factorising-q3

Show solution

Worked solution

1.Spot the skill: Look for two numbers that MULTIPLY to c and ADD to b in x² + bx + c.
2.Use the write the factorised form stage first, then apply the zero-product property.
3.Keep the final answer visible: x = 3 or x = 6.

Final answer

x = 3 or x = 6

Mark points

M1: use the correct look for two numbers that multiply to c and add to b in x² + bx + c.write as (x + p)(x + q) = 0, then set each bracket to zero.for 2ax² + bx + c: multiply a by c, find two numbers multiplying to ac and adding to b, split the middle term.
A1: x = 3 or x = 6

Watch out

g. 1 × 6 = 6 but 1 + 6 = 7 ≠ 5). Always write the pair check explicitly.Also, ensure the equation equals zero — rearranging first is essential if it does not.

Hardproblem solving

Solve x² − 25 = 0 by factorising (difference of two squares)

3 marks5 minssolving-quadratics-by-factorising-q4

Show solution

Worked solution

1.Spot the skill: Look for two numbers that MULTIPLY to c and ADD to b in x² + bx + c.
2.Use the apply the zero-product property stage first, then make sure one side equals zero.
3.Keep the final answer visible: x = 5 or x = −5.

Final answer

x = 5 or x = −5

Mark points

M1: use the correct look for two numbers that multiply to c and add to b in x² + bx + c.write as (x + p)(x + q) = 0, then set each bracket to zero.for 2ax² + bx + c: multiply a by c, find two numbers multiplying to ac and adding to b, split the middle term.
A1: x = 5 or x = −5

Watch out

g. 1 × 6 = 6 but 1 + 6 = 7 ≠ 5). Always write the pair check explicitly.Also, ensure the equation equals zero — rearranging first is essential if it does not.

Exam-stylemulti-step

Solve 2x² + 7x + 3 = 0

4 marks6 minssolving-quadratics-by-factorising-q5

Show solution

Worked solution

1.Spot the skill: Look for two numbers that MULTIPLY to c and ADD to b in x² + bx + c.
2.Use the make sure one side equals zero stage first, then find a pair of numbers that multiply to 6 and add to 5.
3.Keep the final answer visible: x = − $\frac{1}{2}$ or x = −3.

Final answer

x = − $\frac{1}{2}$ or x = −3

Mark points

M1: use the correct look for two numbers that multiply to c and add to b in x² + bx + c.write as (x + p)(x + q) = 0, then set each bracket to zero.for 2ax² + bx + c: multiply a by c, find two numbers multiplying to ac and adding to b, split the middle term.
A1: x = − $\frac{1}{2}$ or x = −3

Watch out

g. 1 × 6 = 6 but 1 + 6 = 7 ≠ 5). Always write the pair check explicitly.Also, ensure the equation equals zero — rearranging first is essential if it does not.

Grade 9 stretchproblem solving

Solve 2x² - 7x + 3 = 0.

4 marks7 minsquad-factor-g9

Show solution

Worked solution

1.Find factors that create the middle term.
2.Factorise as two brackets.
3.Set each bracket equal to zero.

Final answer

x = 3 or x = $\frac{1}{2}$

Mark points

M1: (2x - 1)(x - 3)
A1: both solutions

Watch out

Do not rush straight into arithmetic. Select the relevant method and show a complete chain of working.

Timed checkpoint

16 mins - 9 marks

Switch between skills

Set a timer and attempt all four questions before opening any answers. This is closer to the way skills appear in a real paper.

1Solving quadratics by factorising - 2 marksSolve x² + 7x + 12 = 0Mark answer

Answer

x = −3 or x = −4

2Algebraic notation and substitution - 2 marksFind t² + 4t when t = −2Mark answer

Answer

−4

3Simplifying expressions - 2 marksSimplify 2xy + 3x²y − xy + x²yMark answer

Answer

3x²y + xy

4Expanding and factorising - 3 marksFactorise fully 6x² + 9xMark answer

Answer

3x(2x + 3)

Mastery check

I can explain the method for solving quadratics by factorising.
I can show clear working without skipping key steps.
g. 1 × 6 = 6 but 1 + 6 = 7 ≠ 5). Always write the pair check explicitly.Also, ensure the equation equals zero — rearranging first is essential if it does not.

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Factorise, then set each bracket to zero

What you will learn

Know the rule, then use it

Method

Make sure one side equals zero

Find a pair of numbers that multiply to 6 and add to 5

Write the factorised form

Watch out

Solve x2 + 5x + 6 = 0 by factorising.

Build up to the hardest questions

Switch between skills

Get help with anything that still feels tricky.

Solve x² + 5x + 6 = 0 by factorising.