Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 472 · 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.What are the roots of x² − 11x + 30 = 0? (a) 1 and 10 (b) 2 and 15 (c) 1 and 30 (d) 5 and 6
2. 2.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 7056 (b) 5786 (c) 6384 (d) 5776
3. 3.What is the product of the roots of x² − 10x + 21 = 0? (a) 10 (b) 22 (c) 20 (d) 21
4. 4.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 896 (b) 794 (c) 784 (d) 1024
5. 5.What is the sum of the roots of x² − 13x + 36 = 0? (a) 36 (b) 14 (c) 12 (d) 13
6. 6.What are the roots of x² − 10x + 21 = 0? (a) 3 and 7 (b) 2 and 8 (c) 1 and 9 (d) 1 and 21
7. 7.If a + b = 16 and ab = 63, what is a² + b²? (a) 130 (b) 193 (c) 256 (d) 382
8. 8.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2611 (b) 2501 (c) 2601 (d) 2401
9. 9.If a + b = 17 and ab = 70, what is (a − b)²? (a) 289 (b) 3 (c) 9 (d) 149
10. 10.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3584 (b) 4096 (c) 3136 (d) 3146
11. 11.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 7569 (b) 8109 (c) 8659 (d) 8649
12. 12.If a + b = 8 and ab = 12, what is a² + b²? (a) 52 (b) 88 (c) 64 (d) 40
13. 13.If a + b = 5 and ab = 4, what is (a − b)²? (a) 9 (b) 3 (c) 25 (d) 17
14. 14.If a + b = 19 and ab = 88, what is a² + b²? (a) 273 (b) 537 (c) 361 (d) 185
15. 15.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 1)(x + 36) (c) (x + 1)(x + 12) (d) (x + 4)(x + 9)
16. 16.Which is the correct factorisation of x² + 7x + 10? (a) (x − 2)(x − 5) (b) (x + 1)(x + 6) (c) (x + 2)(x + 5) (d) (x + 1)(x + 10)
17. 17.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1681 (b) 1691 (c) 1521 (d) 1601
18. 18.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4624 (b) 5194 (c) 5184 (d) 4904
19. 19.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10404 (b) 10004 (c) 9604 (d) 10414
20. 20.What is the product of the roots of x² − 7x + 12 = 0? (a) 13 (b) 12 (c) 11 (d) 7
21. 21.If a + b = 11 and ab = 30, what is a² + b²? (a) 61 (b) 121 (c) 181 (d) 91
22. 22.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x − 3)(x − 4) (c) (x + 1)(x + 12) (d) (x + 3)(x + 4)
23. 23.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2126 (b) 2916 (c) 2116 (d) 2484
24. 24.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
25. 25.Which is the correct factorisation of x² + 9x + 18? (a) (x + 3)(x + 6) (b) (x + 1)(x + 8) (c) (x + 1)(x + 18) (d) (x − 3)(x − 6)
26. 26.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9035 (b) 11025 (c) 9025 (d) 9975
27. 27.If a + b = 12 and ab = 20, what is a² + b²? (a) 144 (b) 124 (c) 184 (d) 104
28. 28.If a + b = 13 and ab = 12, what is a² + b²? (a) 157 (b) 193 (c) 145 (d) 169
29. 29.If a + b = 18 and ab = 72, what is a² + b²? (a) 252 (b) 468 (c) 180 (d) 324
30. 30.What are the roots of x² − 14x + 48 = 0? (a) 2 and 24 (b) 1 and 13 (c) 1 and 48 (d) 6 and 8
31. 31.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 8836 (b) 8084 (c) 7406 (d) 7396
32. 32.What is the product of the roots of x² − 11x + 30 = 0? (a) 11 (b) 31 (c) 29 (d) 30
33. 33.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 63) (c) (x + 7)(x + 9) (d) (x + 1)(x + 15)
34. 34.If a + b = 16 and ab = 55, what is a² + b²? (a) 366 (b) 256 (c) 146 (d) 201
35. 35.If a + b = 15 and ab = 54, what is a² + b²? (a) 333 (b) 117 (c) 171 (d) 225
36. 36.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8281 (b) 7921 (c) 8099 (d) 7931
37. 37.What is the sum of the roots of x² − 11x + 28 = 0? (a) 28 (b) 11 (c) 10 (d) 12
38. 38.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1764 (b) 1444 (c) 1604 (d) 1774
39. 39.If a + b = 7 and ab = 10, what is (a − b)²? (a) 9 (b) 49 (c) 29 (d) 3
40. 40.Which is the correct factorisation of x² + 8x + 12? (a) (x − 2)(x − 6) (b) (x + 2)(x + 6) (c) (x + 1)(x + 12) (d) (x + 1)(x + 7)

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