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 Expand (x + 6)(x − 4).
Show worked solution
x2 − 4x + 6x − 24 = x2 + 2x − 24. x2 − 24 forgets the middle terms; x2 − 2x − 24 gets the sign of the middle term wrong; x2 + 10x − 24 adds 6 and 4 and ignores the minus sign.
2 Expand (3x + 2)(x + 5).
Show worked solution
3x2 + 15x + 2x + 10 = 3x2 + 17x + 10.
3 Expand (x − 9)(x + 2). What is the coefficient of x?
Show worked solution
x2 + 2x − 9x − 18 = x2 − 7x − 18, so the coefficient of x is −7.
Level 2 · Core
4 Expand (2y − 5)(3y − 2).
Show worked solution
6y2 − 4y − 15y + 10 = 6y2 − 19y + 10. 5y2 adds the coefficients 2 + 3 instead of multiplying; −10 is the sign error −5 × −2 = −10; +19y loses the minus signs on the middle terms.
5 Expand and simplify (4a − 3)(2a + 7) − 5a2.
Show worked solution
8a2 + 28a − 6a − 21 = 8a2 + 22a − 21, and 8a2 − 5a2 = 3a2. Answer: 3a2 + 22a + (−21), that is 3a2 + 22a − 21.
6 A rectangle is (3n + 2) cm long and (n − 1) cm wide. Its area, expanded, is 3n2 + bn − 2. Find b.
Show worked solution
(3n + 2)(n − 1) = 3n2 − 3n + 2n − 2 = 3n2 − n − 2, so b = −1.
Level 3 · Challenge
7 Three statements:
1 (x − 3)(x + 8) = x2 − 5x − 24
2 (2k + 3)(2k − 5) = 4k2 − 4k − 15
3 (1 − 3t)(2 − t) = 3t2 − 7t + 2
Which statements are correct?
1 (x − 3)(x + 8) = x2 − 5x − 24
2 (2k + 3)(2k − 5) = 4k2 − 4k − 15
3 (1 − 3t)(2 − t) = 3t2 − 7t + 2
Which statements are correct?
Show worked solution
8x − 3x = +5x, so (x − 3)(x + 8) = x2 + 5x − 24, not x2 − 5x − 24 (1 ✗). −10k + 6k = −4k and 3 × −5 = −15 (2 ✓). 2 − t − 6t + 3t2 = 3t2 − 7t + 2 (3 ✓). 2 and 3.
8 (ax − 5)(x + b) = 2x2 + x − 15 for every x. Find the numbers a and b.
Show worked solution
The left side is ax2 + (ab − 5)x − 5b. Matching x2: a = 2. Matching numbers: −5b = −15, so b = 3. Check: ab − 5 = 6 − 5 = 1 ✓.
9 Expand and simplify (3x − 2)(x + 6) − (x − 5)(3x + 1). What is the coefficient of x?
Show worked solution
(3x − 2)(x + 6) = 3x2 + 16x − 12 and (x − 5)(3x + 1) = 3x2 − 14x − 5. Subtracting: 16x − (−14x) = 30x, and the whole result is 30x − 7. The coefficient of x is 30.