Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 115 · 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.If a + b = 8 and ab = 15, what is (a − b)²? (a) 34 (b) 4 (c) 2 (d) 64
2. 2.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 8125 (b) 7225 (c) 9025 (d) 9035
3. 3.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 7921 (b) 8281 (c) 8101 (d) 8291
4. 4.If a + b = 19 and ab = 88, what is a² + b²? (a) 537 (b) 273 (c) 185 (d) 361
5. 5.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4904 (b) 4624 (c) 5194 (d) 5184
6. 6.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10009 (b) 9409 (c) 10609 (d) 10619
7. 7.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 375 (b) 225 (c) 625 (d) 235
8. 8.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 324 (b) 404 (c) 494 (d) 484
9. 9.What is the product of the roots of x² − 14x + 48 = 0? (a) 49 (b) 14 (c) 48 (d) 47
10. 10.If a + b = 14 and ab = 48, what is a² + b²? (a) 292 (b) 148 (c) 196 (d) 100
11. 11.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3259 (b) 3249 (c) 3591 (d) 3969
12. 12.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 6)(x + 7) (c) (x − 6)(x − 7) (d) (x + 1)(x + 12)
13. 13.If a + b = 6 and ab = 8, what is a² + b²? (a) 36 (b) 28 (c) 52 (d) 20
14. 14.If a + b = 18 and ab = 80, what is (a − b)²? (a) 2 (b) 4 (c) 324 (d) 164
15. 15.What is the sum of the roots of x² − 11x + 28 = 0? (a) 12 (b) 28 (c) 10 (d) 11
16. 16.What is the product of the roots of x² − 16x + 63 = 0? (a) 62 (b) 16 (c) 64 (d) 63
17. 17.If a + b = 14 and ab = 40, what is (a − b)²? (a) 116 (b) 196 (c) 6 (d) 36
18. 18.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1764 (b) 1444 (c) 1774 (d) 1604
19. 19.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2126 (b) 2116 (c) 2484 (d) 2916
20. 20.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4884 (b) 5476 (c) 4356 (d) 4366
21. 21.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 71 (c) 73 (d) 17
22. 22.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2025 (b) 2035 (c) 1625 (d) 1225
23. 23.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 625 (b) 1235 (c) 1225 (d) 925
24. 24.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 16)
25. 25.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7235 (b) 6425 (c) 5625 (d) 7225
26. 26.If a + b = 18 and ab = 72, what is a² + b²? (a) 324 (b) 468 (c) 252 (d) 180
27. 27.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6899 (b) 5929 (c) 6889 (d) 6409
28. 28.If a + b = 9 and ab = 14, what is a² + b²? (a) 81 (b) 67 (c) 53 (d) 109
29. 29.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 5329 (b) 4891 (c) 4489 (d) 4499
30. 30.If a + b = 12 and ab = 27, what is a² + b²? (a) 144 (b) 117 (c) 198 (d) 90
31. 31.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 7569 (b) 8649 (c) 7579 (d) 8091
32. 32.If a + b = 19 and ab = 90, what is a² + b²? (a) 181 (b) 541 (c) 361 (d) 271
33. 33.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x − 3)(x − 5) (c) (x + 1)(x + 15) (d) (x + 3)(x + 5)
34. 34.If a + b = 6 and ab = 8, what is (a − b)²? (a) 20 (b) 36 (c) 4 (d) 2
35. 35.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3035 (b) 4225 (c) 3025 (d) 3575
36. 36.What is the product of the roots of x² − 8x + 15 = 0? (a) 15 (b) 16 (c) 8 (d) 14
37. 37.What are the roots of x² − 12x + 27 = 0? (a) 3 and 9 (b) 1 and 11 (c) 1 and 27 (d) 2 and 10
38. 38.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 15 (c) 14 (d) 13
39. 39.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 8) (b) (x − 2)(x − 7) (c) (x + 2)(x + 7) (d) (x + 1)(x + 14)
40. 40.If a + b = 9 and ab = 18, what is (a − b)²? (a) 9 (b) 45 (c) 3 (d) 81

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