What this guide covers
Sec 2 N5 quadratic factorisation and algebraic identities. The Upper Secondary guide continues into equation and error-analysis applications.
Choose a route
First extract any factor shared by every term. For a quadratic, find two products that reproduce both the middle term and constant. For a difference of squares, use A²−B²=(A−B)(A+B).
Factor a non-monic quadratic
Factorise 6x²+7x−3 completely.
- Split 7x into 9x−2x because 9×(−2)=−18=6×(−3).
- 6x²+9x−2x−3=3x(2x+3)−1(2x+3).
- The result is (3x−1)(2x+3).
Check: Expanding gives 6x²+9x−2x−3=6x²+7x−3.
Remove a common factor first
Factorise 3x²−12x+9 completely.
- Take out 3: 3(x²−4x+3).
- The inner quadratic is (x−1)(x−3).
- The complete factorisation is 3(x−1)(x−3).
Check: 3(x²−4x+3)=3x²−12x+9.
Expand to verify
Multiplying the proposed factors must return every term of the original expression, including its signs.
Try it yourself
Work on paper, then compare the steps. These checks do not establish mastery.
Factorise x²−7x+12.
(x−3)(x−4); the numbers −3 and −4 multiply to 12 and sum to −7.
Factorise 16p²−25.
(4p−5)(4p+5), a difference of squares.
Factorise 2y²+5y+2 and verify by expansion.
(2y+1)(y+2). Expansion gives 2y²+4y+y+2=2y²+5y+2.
Where this connects
- Earlier: expand brackets →
Expansion is the inverse check for every factorisation here.
- Later: factorisation in equations →
The Upper Secondary guide uses the same structure for a stronger error-analysis problem.
Sources and scope
Original Mentora examples. Sec 2 N5 quadratic factorisation is in the MOE G2/G3 Mathematics syllabus and appears in the previously sampled Broadrick Sec 2 assessment; no paper wording is reproduced.