Quadratics and the zero-product rule
Factor a quadratic and use its product form to find roots.
Before you begin
- Expanding brackets
- Signed-number multiplication
Reverse expansion
Expanding (x + a)(x + b) gives x² + (a + b)x + ab. To factor a monic quadratic, look for two numbers whose sum is the coefficient of x and whose product is the constant term. Both conditions must hold. Check the proposed factors by expanding them before solving an equation.
Set each factor equal to zero
If a product is zero, at least one factor is zero. This allows (x + a)(x + b) = 0 to become two simple equations. First rearrange the original equation so one side is zero. The rule does not say that either factor equals a nonzero right-hand side.
Worked example
Solve x² − 5x + 6 = 0.
Show the worked solution
- Find numbers with sum −5 and product 6: −2 and −3.
- Factor to (x − 2)(x − 3) = 0.
- Set each factor to zero and check both roots in the original equation.
x = 2 or x = 3.
Try it yourself
Which is the factorisation of x² + 7x + 12?
Common mistakes
- A quadratic can have two solutions; do not stop after one.
What can you explain now?
Solve x² + x − 12 = 0.
Compare with the explanation
x = 3 or x = −4.
The numbers 4 and −3 have sum 1 and product −12. Thus (x + 4)(x − 3) = 0, giving both roots.
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Original StudyForA content. AI-assisted checks completed; subject-teacher review is still pending. These lessons are not endorsed by an exam board.
Common Core mathematics standards ↗