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 Solve x2 − 10x + 21 = 0.
Show worked solution
−3 × −7 = 21 and −3 + −7 = −10, so (x − 3)(x − 7) = 0 and x = 3 or x = 7. Check: 3 + 7 = 10 = −b. x = −7 or x = −3 copies the bracket numbers without flipping the signs; x = −7 or x = 3 and x = −3 or x = 7 come from factor pairs that multiply to −21.
2 Solve x2 + 9x + 18 = 0.
Show worked solution
3 × 6 = 18 and 3 + 6 = 9, so (x + 3)(x + 6) = 0, giving x = −3 or x = −6. Smaller root −6, larger root −3. Check: −6 + −3 = −9 = −b and (−6)(−3) = 18 = c.
3 Solve x2 + 4x = 45. What is the sum of the two solutions?
Show worked solution
x2 + 4x − 45 = 0, so (x + 9)(x − 5) = 0 and x = −9 or x = 5. Sum: −9 + 5 = −4. Check: the sum is −b = −4 ✓.
Level 2 · Core
4 Solve x2 + 6x = 16.
Show worked solution
x2 + 6x − 16 = 0, so (x + 8)(x − 2) = 0 and x = −8 or x = 2. Check: 64 − 48 = 16 ✓ and 4 + 12 = 16 ✓. x = 16 or x = 10 writes x(x + 6) = 16 and sets each factor to 16; x = 0 or x = −6 ignores the 16; x = −2 or x = 8 flips both signs.
5 Solve 2x2 − 8x − 42 = 0.
Show worked solution
2(x2 − 4x − 21) = 0, so 2(x − 7)(x + 3) = 0 and x = 7 or x = −3. Smaller root −3, larger root 7. Check: 2 × 9 + 24 − 42 = 0 ✓ and 2 × 49 − 56 − 42 = 0 ✓.
6 The parabola is y = x2 + bx + c. Use the points where it crosses the x-axis to find b.
Show worked solution
The roots are x = −3 and x = 1, so y = (x + 3)(x − 1) = x2 + 2x − 3 and b = 2. Check: the roots add to −2 = −b ✓.
Level 3 · Challenge
7 Three statements about the equation 2x2 = 14x:
1 Dividing both sides by 2x gives every solution.
2 x = 0 is a solution.
3 The solutions add to 7.
Which statements are correct?
1 Dividing both sides by 2x gives every solution.
2 x = 0 is a solution.
3 The solutions add to 7.
Which statements are correct?
Show worked solution
2x2 − 14x = 0, so 2x(x − 7) = 0 and x = 0 or x = 7. Dividing by 2x gives only x = 7 and loses x = 0 (1 ✗). 2 × 02 = 0 = 14 × 0 (2 ✓). 0 + 7 = 7 (3 ✓). 2 and 3.
8 Solve (x + 1)(x − 4) = 14.
Show worked solution
x2 − 3x − 4 = 14, so x2 − 3x − 18 = 0 and (x − 6)(x + 3) = 0. Smaller root −3, larger root 6. Check: (−2)(−7) = 14 ✓ and 7 × 2 = 14 ✓. Setting each bracket equal to 14 (x = 13 or x = 18) is the forgot-to-set-0 error.
9 One solution of x2 + kx − 42 = 0 is x = 6. Find k.
Show worked solution
The roots multiply to −42, so the other root is −42 ÷ 6 = −7. The roots add to 6 + (−7) = −1 = −k, so k = 1. Check: x2 + x − 42 = (x + 7)(x − 6) ✓, and 36 + 6 − 42 = 0 ✓.