Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 721 · 40 questions

How to solve it (Use the identity, skip the guessing): (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b²; so a² + b² = (a + b)² − 2ab and (a − b)² = (a + b)² − 4ab. For x² − Sx + P = 0: the sum of the roots is S and the product is P. To factorise x² + Sx + P, find two numbers that multiply to P and add up to S.

1. 1.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 4096 (b) 3146 (c) 3584 (d) 3136
2. 2.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4489 (b) 4499 (c) 5329 (d) 4891
3. 3.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 17 (c) 16 (d) 72
4. 4.What is the sum of the roots of x² − 12x + 35 = 0? (a) 11 (b) 12 (c) 35 (d) 13
5. 5.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 7921 (b) 8281 (c) 8099 (d) 7931
6. 6.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1225 (b) 625 (c) 925 (d) 1235
7. 7.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 2)(x + 4) (c) (x − 2)(x − 4) (d) (x + 1)(x + 5)
8. 8.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 73 (c) 71 (d) 72
9. 9.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8649 (b) 8659 (c) 7569 (d) 8109
10. 10.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 1 and 16 (c) 8 and 9 (d) 2 and 36
11. 11.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 12) (c) (x + 3)(x + 4) (d) (x + 1)(x + 6)
12. 12.If a + b = 18 and ab = 80, what is (a − b)²? (a) 164 (b) 324 (c) 4 (d) 2
13. 13.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3844 (b) 3596 (c) 3374 (d) 3364
14. 14.What are the roots of x² − 12x + 35 = 0? (a) 1 and 11 (b) 1 and 35 (c) 5 and 7 (d) 2 and 10
15. 15.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2314 (b) 2496 (c) 2304 (d) 2704
16. 16.What is the sum of the roots of x² − 10x + 24 = 0? (a) 24 (b) 11 (c) 9 (d) 10
17. 17.If a + b = 14 and ab = 48, what is (a − b)²? (a) 100 (b) 4 (c) 2 (d) 196
18. 18.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x − 4)(x − 8) (c) (x + 1)(x + 11) (d) (x + 4)(x + 8)
19. 19.What is the sum of the roots of x² − 10x + 21 = 0? (a) 11 (b) 9 (c) 10 (d) 21
20. 20.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7235 (b) 7225 (c) 6425 (d) 5625
21. 21.What are the roots of x² − 7x + 12 = 0? (a) 3 and 4 (b) 1 and 12 (c) 2 and 6 (d) 1 and 6
22. 22.If a + b = 17 and ab = 72, what is (a − b)²? (a) 1 (b) 289 (c) 217 (d) 145
23. 23.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 576 (b) 586 (c) 256 (d) 416
24. 24.What is the sum of the roots of x² − 9x + 20 = 0? (a) 8 (b) 9 (c) 10 (d) 20
25. 25.If a + b = 17 and ab = 66, what is (a − b)²? (a) 25 (b) 5 (c) 157 (d) 289
26. 26.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 676 (b) 884 (c) 686 (d) 1156
27. 27.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 4909 (b) 5339 (c) 5329 (d) 4489
28. 28.If a + b = 5 and ab = 4, what is (a − b)²? (a) 17 (b) 9 (c) 3 (d) 25
29. 29.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1034 (b) 1024 (c) 904 (d) 784
30. 30.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1379 (b) 1591 (c) 1369 (d) 1849
31. 31.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x − 4)(x − 9) (c) (x + 4)(x + 9) (d) (x + 1)(x + 36)
32. 32.If a + b = 12 and ab = 35, what is a² + b²? (a) 144 (b) 214 (c) 74 (d) 109
33. 33.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 4)(x + 7) (c) (x + 1)(x + 10) (d) (x + 1)(x + 28)
34. 34.What is the product of the roots of x² − 9x + 18 = 0? (a) 19 (b) 18 (c) 9 (d) 17
35. 35.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 9409 (b) 10609 (c) 10619 (d) 10009
36. 36.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 8)(x + 9) (d) (x + 1)(x + 16)
37. 37.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 301 (c) 202 (d) 400
38. 38.If a + b = 20 and ab = 96, what is (a − b)²? (a) 16 (b) 400 (c) 208 (d) 4
39. 39.What is the product of the roots of x² − 15x + 56 = 0? (a) 56 (b) 57 (c) 15 (d) 55
40. 40.If a + b = 15 and ab = 36, what is a² + b²? (a) 189 (b) 297 (c) 225 (d) 153

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