Factoring Quadratics
Factoring as Reverse Expansion
Expanding multiplies brackets to create an expression. Factoring works backward: it starts with the expression and recovers the brackets that produced it.
How expansion connects to factoring
When (x + 3)(x + 4) is expanded, the constant terms multiply to 12 and add to make the middle coefficient 7. Factoring x² + 7x + 12 asks you to recover 3 and 4.
Reverse the expansion
- Read the constant term.
- Find two numbers that multiply to that constant.
- Check that the same numbers add to the middle coefficient.
- Use those numbers in two brackets.
Worked example
Factor x² + 7x + 12
- 3 × 4 = 12
- 3 + 4 = 7
- Use +3 and +4 in the brackets.
x² + 7x + 12 = (x + 3)(x + 4).
Common mistakes
- Checking the product but not the sum.
- Forgetting that expanding your answer is a quick way to verify it.
Practice more
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