Success Homeschool

Secondary 4 Mathematics · Factorising quadratic expressions

Name: ______________________

Date: ____________

1. Factorise x^2 + 8x + 15.
- ☐ (a) (x + 3)(x + 5)
- ☐ (b) (x + 1)(x + 15)
- ☐ (c) (x + 2)(x + 6)
- ☐ (d) (x − 3)(x − 5)
2. Factorise x^2 − 49.
- ☐ (a) (x − 7)^2
- ☐ (b) (x + 7)^2
- ☐ (c) (x − 7)(x + 7)
- ☐ (d) (x − 49)(x + 1)
3. Factorise x^2 − 10x + 25.
- ☐ (a) (x + 5)^2
- ☐ (b) (x − 5)(x + 5)
- ☐ (c) (x − 25)(x − 1)
- ☐ (d) (x − 5)^2
4. Factorise x^2 + 2x − 15.
5. Factorise 3x^2 + 12x.
6. Factorise 2x^2 + 5x + 3.
- ☐ (a) (2x + 1)(x + 3)
- ☐ (b) (2x + 3)(x + 1)
- ☐ (c) (2x + 3)(x + 3)
- ☐ (d) (x + 3)(x + 1)
7. Factorise fully 2x^2 − 18.
8. Which expression is a perfect square?
- ☐ (a) x^2 + 6x + 6
- ☐ (b) x^2 + 6x + 9
- ☐ (c) x^2 + 9
- ☐ (d) x^2 − 6x − 9

Answer sheet

1. (x + 3)(x + 5) — 3 × 5 = 15 and 3 + 5 = 8, so x^2 + 8x + 15 = (x + 3)(x + 5).
2. (x − 7)(x + 7) — x^2 − 49 is a difference of two squares: x^2 − 7^2 = (x − 7)(x + 7).
3. (x − 5)^2 — −5 × −5 = 25 and −5 + −5 = −10, so it is a perfect square: (x − 5)^2.
4. (x + 5)(x − 3) — 5 × (−3) = −15 and 5 + (−3) = 2, so x^2 + 2x − 15 = (x + 5)(x − 3).
5. 3x(x + 4) — The highest common factor of 3x^2 and 12x is 3x, leaving x + 4 in the bracket.
6. (2x + 3)(x + 1) — ac = 6; 3 and 2 multiply to 6 and add to 5. 2x^2 + 2x + 3x + 3 = 2x(x + 1) + 3(x + 1) = (2x + 3)(x + 1).
7. 2(x − 3)(x + 3) — Take out 2 to get 2(x^2 − 9), then factorise the difference of two squares: 2(x − 3)(x + 3).
8. x^2 + 6x + 9 — x^2 + 6x + 9 = (x + 3)^2, because 3 × 3 = 9 and 3 + 3 = 6.