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.
15 questions · official format, shuffled order
1 Factorise x2 + 3x − 40.
Show worked solution
8 × (−5) = −40 and 8 + (−5) = 3, so (x + 8)(x − 5). ((x − 8)(x + 5) has the signs swapped: it adds to −3.)
2 Solve x2 − 11x + 24 = 0.
Show worked solution
(x − 3)(x − 8) = 0, so x = 3 or x = 8. (x = −3 or x = −8 reads the numbers straight out of the brackets without flipping the sign.)
3 Solve 3x2 = 18x.
Show worked solution
3x2 − 18x = 0, so 3x(x − 6) = 0: x = 0 or x = 6. (Dividing both sides by x gives x = 6 only and loses x = 0.)
4 Factorise 3x2 + 6x − 72 fully.
Show worked solution
3(x2 + 2x − 24) = 3(x + 6)(x − 4). ((3x − 12)(x + 6) has the right value but is not fully factorised — 3 still divides 3x − 12. (x + 6)(x − 4) has lost the 3.)
5 Two consecutive even positive integers have a product of 224. What is the larger one?
Show worked solution
(n + 16)(n − 14) = 0, so n = 14 (n = −16 is not positive) and the larger is 16. (30 is their sum.)
6 A rectangle is x cm wide and (x + 5) cm long. Its area is 84 cm2. What is its perimeter?
Show worked solution
(x + 12)(x − 7) = 0, so x = 7 and the rectangle is 7 by 12: perimeter 38 cm. (19 is half the perimeter.)
7 Solve 2x3 − 5 = 123.
Show worked solution
2x3 = 128, x3 = 64, x = 4. (8 takes the square root of 64 instead of the cube root.)
8 Solve (3x + 1)3 = −8.
Show worked solution
3x = −3, so x = −1. (−3 skips the cube root and solves 3x + 1 = −8.)
9 Which quadratic equation has the solutions x = −6 and x = 5?
Show worked solution
(x + 6)(x − 5) = x2 + x − 30 = 0. (x2 − x − 30 = 0 has solutions 6 and −5.)
10 The graph of y = x2 − 6x + 5 is shown. How far apart are the two points where it crosses the x-axis?
Show worked solution
(x − 1)(x − 5) = 0 gives x = 1 and x = 5, which are 4 units apart. (6 is their sum; 5 is the y-intercept.)
11 Three statements about the equation x2 = 9x:
1 x = 9 is a solution.
2 x = 0 is a solution.
3 Dividing both sides by x finds every solution.
Which statements are correct?
1 x = 9 is a solution.
2 x = 0 is a solution.
3 Dividing both sides by x finds every solution.
Which statements are correct?
Show worked solution
x(x − 9) = 0 gives 0 and 9 (1 ✓, 2 ✓). Dividing by x throws away x = 0 (3 ✗). 1 and 2.
12 A square garden of side 10 m has a path of width x m all the way round it. The garden and path together cover 196 m2. How wide is the path?
Show worked solution
10 + 2x = 14 (it cannot be −14), so x = 2 m. (Adding x only once gives 10 + x = 14, x = 4, which is not offered — and would be wrong.)
13 When each side of a cube is increased by 4 cm, its volume becomes 1331 cm3. What was the original volume?
Show worked solution
New side 11 cm, old side 7 cm, old volume 73 = 343 cm3. (1327 cm3 takes 4 off the volume instead of the side.)
14 The equation x2 + bx + c = 0 has solutions x = 3 and x = −8. What is b?
Show worked solution
(x − 3)(x + 8) = x2 + 5x − 24, so b = 5. (−5 adds the solutions instead of the numbers in the factors.)
15 Three statements about x2 − 64 = 0:
1 Its solutions are x = 8 and x = −8.
2 x2 − 64 = (x − 8)2.
3 It has the same solutions as x2 = 64.
Which statements are correct?
1 Its solutions are x = 8 and x = −8.
2 x2 − 64 = (x − 8)2.
3 It has the same solutions as x2 = 64.
Which statements are correct?
Show worked solution
(x − 8)(x + 8) = 0 gives ±8 (1 ✓). (x − 8)2 = x2 − 16x + 64 (2 ✗). Adding 64 to both sides gives x2 = 64 (3 ✓). 1 and 3.