Solving Equations by Factorisation (Product = 0)

The zero-product rule, solving equations of the form (x + a)(x + b) = 0, factorising first when the equation is not yet in that form, and dividing polynomials by a monomial.

Everything so far had one answer. Once an equation contains an x² it can have two, and the way in is a single idea about zero.

1The zero-product rule

If A × B = 0, then A = 0 or B = 0.

If two things multiply to give zero, at least one of them must be zero — there is no other way to reach 0. This is the only rule you need.

⚠️

It only works for zero. If (x + 1)(x + 2) = 6 you may not say x + 1 = 6 or x + 2 = 6. Move everything to one side first so the other side is 0.

2When it is already factorised

Worked ExampleSolve: (x − 3)(x + 5) = 0

1
Set each bracket equal to zero: x − 3 = 0 or x + 5 = 0.
The zero-product rule.
2
x = 3 or x = −5.
Solve each small equation.
3
Check: (3−3)(3+5) = 0 × 8 = 0 ✓
Substituting back confirms both roots.

3Factorise first

Worked ExampleSolve: x² + 7x + 12 = 0

1
The right side is already 0, so factorise the left.
2
Two numbers multiplying to 12 and adding to 7: 3 and 4.
x² + 7x + 12 = (x + 3)(x + 4).
3
(x + 3)(x + 4) = 0, so x = −3 or x = −4.
Note the signs flip when you solve.

Worked ExampleSolve: x² − 5x = 0

1
Take out the common factor: x(x − 5) = 0.
Common factor first, always.
2
x = 0 or x − 5 = 0.
3
x = 0 or x = 5.
Do NOT divide both sides by x — you would lose the root x = 0.

⚠️

Never divide an equation by x. Dividing x² = 5x by x gives only x = 5 and silently throws away x = 0. Always move everything to one side and factorise instead.

Worked ExampleSolve: x² − 16 = 0

1
Recognise a difference of two squares.
x² − 16 = x² − 4².
2
(x − 4)(x + 4) = 0.
3
x = 4 or x = −4.

4Dividing a polynomial by a monomial

The Grade 9 ATP also asks you to divide a whole expression by a single term. Divide every term separately.

Worked ExampleSimplify: (12x³ − 8x² + 4x) ÷ 4x

1
Split the division across each term.
Each term is divided by 4x on its own.
2
12x³ ÷ 4x = 3x²; −8x² ÷ 4x = −2x; 4x ÷ 4x = 1.
Divide the numbers, subtract the exponents.
3
Answer: 3x² − 2x + 1.
The last term gives 1, not 0 — a very common slip.

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
Solve: (x − 2)(x + 6) = 0
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x = 2 or x = −6
2
Solve: (x + 1)(x − 7) = 0
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x = −1 or x = 7
3
Solve: x² + 5x + 6 = 0
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(x+2)(x+3) = 0 → x = −2 or x = −3
4
Solve: x² − 7x + 10 = 0
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(x−2)(x−5) = 0 → x = 2 or x = 5
5
Solve: x² − 3x = 0
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x(x−3) = 0 → x = 0 or x = 3
6
Solve: x² − 25 = 0
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(x−5)(x+5) = 0 → x = 5 or x = −5
7
Why may you not divide x² = 4x by x?
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You would lose the root x = 0.
8
Simplify: (10x³ + 5x²) ÷ 5x
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2x² + x
9
Simplify: (12x³ − 8x² + 4x) ÷ 4x
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3x² − 2x + 1