Work through the levels in order. Type your answer and press Check — open the worked solution if you get stuck. Answers like -6, 3/4 or 0.25 are typed exactly as shown. A ⭐ Bonus Challenge at the end is extra credit — skip it if the rest was hard work.
Level 1 · Warm-up
1 Factorise x2 − 4x − 32.
Show worked solution
−8 × 4 = −32 and −8 + 4 = −4, so the answer is (x − 8)(x + 4). Check: x2 + 4x − 8x − 32 = x2 − 4x − 32 ✓. (x + 8)(x − 4) swaps the signs and gives +4x; (x − 8)(x − 4) gives +32; (x − 16)(x + 2) multiplies to −32 but adds to −14.
2 x2 − 11x + 28 = (x − a)(x − b), where a and b are positive and a is the smaller. Find a and b.
Show worked solution
Factor pairs of 28: 1 and 28, 2 and 14, 4 and 7. Only 4 + 7 = 11, so x2 − 11x + 28 = (x − 4)(x − 7): a = 4 and b = 7. Check: x2 − 7x − 4x + 28 ✓.
3 x2 + bx + 45 = (x + 5)(x + q). Find b.
Show worked solution
5q = 45, so q = 9. Then b = 5 + 9 = 14. Check: (x + 5)(x + 9) = x2 + 9x + 5x + 45 = x2 + 14x + 45 ✓.
Level 2 · Core
4 Factorise 2x2 + 4x − 70 completely.
Show worked solution
2x2 + 4x − 70 = 2(x2 + 2x − 35). 7 × (−5) = −35 and 7 + (−5) = 2, so the answer is 2(x + 7)(x − 5). 2(x − 7)(x + 5) swaps the signs (middle term −2x); (x + 7)(x − 5) drops the common factor 2; 2(x + 35)(x − 1) multiplies to −35 but adds to 34.
5 x2 + 5x − 36 = (x + m)(x − n), where m and n are positive whole numbers. Find m and n.
Show worked solution
Factor pairs of 36: 1 and 36, 2 and 18, 3 and 12, 4 and 9, 6 and 6. Only 9 and 4 differ by 5, and b is positive, so the bigger one is +9: (x + 9)(x − 4). m = 9 and n = 4. Check: x2 − 4x + 9x − 36 = x2 + 5x − 36 ✓.
6 x2 − 20x + c is a perfect square. Find c.
Show worked solution
Half of −20 is −10, and (−10)2 = 100, so c = 100: x2 − 20x + 100 = (x − 10)2. Check: the middle term is 2 × (−10)x = −20x ✓.
Level 3 · Challenge
7 Three statements:
1 x2 − 3x − 40 = (x − 8)(x + 5)
2 2x2 − 50 = 2(x + 5)(x − 5)
3 x2 + 6x + 36 = (x + 6)2
Which statements are correct?
1 x2 − 3x − 40 = (x − 8)(x + 5)
2 2x2 − 50 = 2(x + 5)(x − 5)
3 x2 + 6x + 36 = (x + 6)2
Which statements are correct?
Show worked solution
(x − 8)(x + 5) = x2 + 5x − 8x − 40 = x2 − 3x − 40 (1 ✓). 2(x2 − 25) = 2x2 − 50 (2 ✓). (x + 6)2 = x2 + 12x + 36: half of 6 is 3, and 32 = 9, not 36, so x2 + 6x + 36 is not a perfect square (3 ✗). 1 and 2.
8 x2 + kx + 24 factorises as (x + p)(x + q), where p and q are whole numbers (they may be negative). Find the smallest and the largest possible values of k.
Show worked solution
Positive pairs: 1 + 24 = 25, 2 + 12 = 14, 3 + 8 = 11, 4 + 6 = 10. The negative pairs give −25, −14, −11 and −10. So the smallest k is −25, from (x − 1)(x − 24), and the largest is 25, from (x + 1)(x + 24).
9 Find the value of a2 − 6a + 9 when a = 53, without a calculator.
Show worked solution
Half of −6 is −3 and (−3)2 = 9, so a2 − 6a + 9 = (a − 3)2. When a = 53: (53 − 3)2 = 502 = 2500.