Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 130 · 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 = 21 and ab = 108, what is a² + b²? (a) 657 (b) 441 (c) 225 (d) 333
2. 2.If a + b = 13 and ab = 30, what is (a − b)²? (a) 169 (b) 109 (c) 49 (d) 7
3. 3.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3249 (b) 3969 (c) 3591 (d) 3259
4. 4.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)
5. 5.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4356 (b) 5476 (c) 5486 (d) 4916
6. 6.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 1024 (b) 896 (c) 784 (d) 794
7. 7.If a + b = 3 and ab = 2, what is a² + b²? (a) 5 (b) 7 (c) 9 (d) 13
8. 8.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x − 5)(x − 8) (c) (x + 1)(x + 40) (d) (x + 5)(x + 8)
9. 9.If a + b = 9 and ab = 20, what is a² + b²? (a) 81 (b) 61 (c) 41 (d) 121
10. 10.What is the sum of the roots of x² − 12x + 35 = 0? (a) 13 (b) 11 (c) 35 (d) 12
11. 11.If a + b = 19 and ab = 88, what is a² + b²? (a) 361 (b) 273 (c) 537 (d) 185
12. 12.What is the sum of the roots of x² − 11x + 30 = 0? (a) 12 (b) 11 (c) 10 (d) 30
13. 13.What is the product of the roots of x² − 6x + 8 = 0? (a) 8 (b) 9 (c) 7 (d) 6
14. 14.If a + b = 18 and ab = 80, what is a² + b²? (a) 484 (b) 324 (c) 164 (d) 244
15. 15.What are the roots of x² − 16x + 63 = 0? (a) 1 and 63 (b) 7 and 9 (c) 1 and 15 (d) 3 and 21
16. 16.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 8125 (b) 9035 (c) 9025 (d) 7225
17. 17.What is the sum of the roots of x² − 9x + 14 = 0? (a) 14 (b) 10 (c) 9 (d) 8
18. 18.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 1)(x + 45) (c) (x + 5)(x + 9) (d) (x + 1)(x + 13)
19. 19.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 18 (c) 16 (d) 72
20. 20.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 5929 (b) 6889 (c) 6409 (d) 6899
21. 21.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x + 1)(x + 8) (c) (x − 3)(x − 6) (d) (x + 3)(x + 6)
22. 22.If a + b = 16 and ab = 60, what is (a − b)²? (a) 4 (b) 136 (c) 16 (d) 256
23. 23.If a + b = 16 and ab = 60, what is a² + b²? (a) 256 (b) 136 (c) 196 (d) 376
24. 24.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1444 (b) 1774 (c) 1604 (d) 1764
25. 25.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x + 1)(x + 10) (c) (x − 3)(x − 8) (d) (x + 3)(x + 8)
26. 26.If a + b = 14 and ab = 48, what is (a − b)²? (a) 4 (b) 196 (c) 2 (d) 100
27. 27.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1691 (b) 1601 (c) 1681 (d) 1521
28. 28.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3364 (b) 3844 (c) 3604 (d) 3854
29. 29.If a + b = 20 and ab = 99, what is (a − b)²? (a) 4 (b) 2 (c) 400 (d) 202
30. 30.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 9604 (b) 10414 (c) 10004 (d) 10404
31. 31.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 15) (c) (x + 7)(x + 9) (d) (x + 1)(x + 63)
32. 32.What is the sum of the roots of x² − 14x + 48 = 0? (a) 13 (b) 15 (c) 48 (d) 14
33. 33.If a + b = 11 and ab = 24, what is a² + b²? (a) 73 (b) 169 (c) 121 (d) 97
34. 34.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 9409 (b) 10609 (c) 10619 (d) 10009
35. 35.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 676 (b) 1166 (c) 1156 (d) 916
36. 36.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 625 (b) 225 (c) 635 (d) 425
37. 37.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 1)(x + 72) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)
38. 38.Which is the correct factorisation of x² + 10x + 21? (a) (x + 3)(x + 7) (b) (x + 1)(x + 21) (c) (x + 1)(x + 9) (d) (x − 3)(x − 7)
39. 39.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 384 (b) 266 (c) 256 (d) 576
40. 40.What are the roots of x² − 13x + 36 = 0? (a) 1 and 36 (b) 4 and 9 (c) 1 and 12 (d) 2 and 18

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