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 and simplify (x + 3)2 − 6x.
Show worked solution
(x + 3)2 = x2 + 6x + 9, and 6x − 6x = 0, leaving x2 + 9. x2 − 6x + 9 is the missing-middle-term error; x2 − 3x + 9 uses only 3x as the middle term; x2 + 12x + 9 adds the 6x instead of subtracting it.
2 Expand (4m − 3)2.
Show worked solution
(4m)2 = 16m2, 2 × 4m × (−3) = −24m, (−3)2 = 9. Answer: 16m2 + (−24)m + 9, that is 16m2 − 24m + 9.
3 Use the difference of two squares to work out 37 × 43.
Show worked solution
402 − 32 = 1600 − 9 = 1591.
Level 2 · Core
4 Expand and simplify (2a + 5b)(2a − 5b) − (2a − 5b)2.
Show worked solution
(4a2 − 25b2) − (4a2 − 20ab + 25b2) = 4a2 − 25b2 − 4a2 + 20ab − 25b2 = 20ab − 50b2. −50b2 is the missing-middle-term error; −20ab subtracts only the first term of the square; 20ab writes the last term of the square as −25b2, but a square is always positive.
5 Expand and simplify (x + 8)2 − (x − 3)(x + 3).
Show worked solution
x2 + 16x + 64 − (x2 − 9) = 16x + 64 + 9 = 16x + 73.
6 Work out 792 by writing it as (80 − 1)2.
Show worked solution
802 − 2 × 80 × 1 + 12 = 6400 − 160 + 1 = 6241.
Level 3 · Challenge
7 Three statements:
1 (x − y)2 = (y − x)2
2 (a + b)2 − (a − b)2 = 4ab
3 522 − 482 = 16
Which statements are correct?
1 (x − y)2 = (y − x)2
2 (a + b)2 − (a − b)2 = 4ab
3 522 − 482 = 16
Which statements are correct?
Show worked solution
Both sides of statement 1 are x2 − 2xy + y2 (1 ✓). a2 + 2ab + b2 − (a2 − 2ab + b2) = 4ab (2 ✓). 522 − 482 = (52 + 48)(52 − 48) = 100 × 4 = 400, not 16 (3 ✗). 1 and 2.
8 (ax − 4)2 = 25x2 + bx + 16 for every x, and a is positive. Find a and b.
Show worked solution
(ax − 4)2 = a2x2 − 8ax + 16. Matching x2: a2 = 25, and a is positive, so a = 5. Then b = −8 × 5 = −40.
9 x + y = 9 and xy = 14. Find the value of x2 + y2 without finding x and y.
Show worked solution
(x + y)2 = x2 + 2xy + y2, so 81 = x2 + y2 + 28 and x2 + y2 = 53. Check: x = 2, y = 7 gives 4 + 49 = 53 ✓.