Expanding and Factorising: Algebra Steps for Year 10–11
Quick answer: Expanding removes brackets by multiplying everything inside by what's outside. Factorising is the reverse: putting an expression back into brackets. For a quadratic like x² + 7x + 12, find two numbers that multiply to 12 and add to 7 (3 and 4), giving (x + 3)(x + 4).
Expanding
One bracket: 3(x + 4) = 3x + 12.
Two brackets: multiply each term in the first by each term in the second.
(x + 3)(x + 5) = x² + 5x + 3x + 15 = x² + 8x + 15.
A square: (x − 4)² = (x − 4)(x − 4) = x² − 8x + 16.
Difference of two squares: (x + 6)(x − 6) = x² − 36.
Factorising
Common factor
Factorise 6x + 9. Both terms divide by 3: 3(2x + 3).
Quadratics where a = 1
Factorise x² + 7x + 12. Multiply to 12, add to 7: 3 and 4. (x + 3)(x + 4).
Factorise x² − x − 20. Multiply to −20, add to −1: −5 and 4. (x − 5)(x + 4).
Difference of two squares
Factorise x² − 49. (x − 7)(x + 7).
Quadratics where a isn't 1
Factorise 2x² + 7x + 3.
Try (2x + 1)(x + 3). Check by expanding: 2x² + 6x + x + 3 = 2x² + 7x + 3 ✓.
Solving by factorising
Solve x² + 7x + 12 = 0.
(x + 3)(x + 4) = 0, so x + 3 = 0 or x + 4 = 0.
x = −3 or x = −4.
Always check
Expand your answer. If you get back to the original expression, the factorising is right.
Common mistakes
Missing the middle term. (x + 3)² is x² + 6x + 9, not x² + 9.
Sign slips. For x² − x − 20, the numbers must multiply to −20.
Not taking out the whole common factor. 6x + 9 = 3(2x + 3), not just partly.
When extra support might help
Support is worth considering when algebra steps get mixed up, or before NCEA assessments.
A tutor can't guarantee a grade, but can make the method reliable. Math Make Smart offers online maths tutoring in New Zealand, and you can book a free trial lesson. See also our simultaneous equations guide and NCEA Level 1 maths standards guide.
Frequently asked questions
What's the difference between expanding and factorising?
Expanding removes brackets; factorising puts them back.
How do I factorise x² + bx + c?
Find two numbers that multiply to c and add to b.
How do I check my factorising?
Expand it and see if you get the original.