Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 923 · 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 55²? (a) 2025 (b) 3035 (c) 3025 (d) 2525
2. 2.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 1)(x + 11) (c) (x + 1)(x + 27) (d) (x + 3)(x + 9)
3. 3.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3146 (b) 3584 (c) 4096 (d) 3136
4. 4.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 63) (c) (x + 1)(x + 15) (d) (x + 7)(x + 9)
5. 5.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1166 (b) 1156 (c) 676 (d) 916
6. 6.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x − 4)(x − 9) (c) (x + 4)(x + 9) (d) (x + 1)(x + 12)
7. 7.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 16 (c) 17 (d) 15
8. 8.If a + b = 13 and ab = 12, what is a² + b²? (a) 169 (b) 145 (c) 193 (d) 157
9. 9.If a + b = 17 and ab = 66, what is a² + b²? (a) 289 (b) 223 (c) 157 (d) 421
10. 10.If a + b = 8 and ab = 12, what is (a − b)²? (a) 4 (b) 40 (c) 64 (d) 16
11. 11.What is the sum of the roots of x² − 13x + 40 = 0? (a) 40 (b) 14 (c) 12 (d) 13
12. 12.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 73 (c) 71 (d) 72
13. 13.What are the roots of x² − 11x + 18 = 0? (a) 1 and 18 (b) 1 and 10 (c) 3 and 6 (d) 2 and 9
14. 14.What are the roots of x² − 14x + 48 = 0? (a) 1 and 48 (b) 6 and 8 (c) 2 and 24 (d) 1 and 13
15. 15.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
16. 16.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x + 1)(x + 11) (c) (x + 5)(x + 7) (d) (x − 5)(x − 7)
17. 17.If a + b = 19 and ab = 88, what is a² + b²? (a) 185 (b) 537 (c) 361 (d) 273
18. 18.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 56) (c) (x + 1)(x + 14) (d) (x − 7)(x − 8)
19. 19.If a + b = 16 and ab = 48, what is (a − b)²? (a) 256 (b) 8 (c) 160 (d) 64
20. 20.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 55 (c) 56 (d) 57
21. 21.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 891 (b) 1089 (c) 739 (d) 729
22. 22.If a + b = 9 and ab = 14, what is (a − b)²? (a) 53 (b) 81 (c) 25 (d) 5
23. 23.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 2025 (b) 1225 (c) 1235 (d) 1575
24. 24.What is the sum of the roots of x² − 8x + 15 = 0? (a) 9 (b) 7 (c) 15 (d) 8
25. 25.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x − 2)(x − 6) (c) (x + 2)(x + 6) (d) (x + 1)(x + 12)
26. 26.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 9) (b) (x − 4)(x − 6) (c) (x + 1)(x + 24) (d) (x + 4)(x + 6)
27. 27.If a + b = 8 and ab = 7, what is a² + b²? (a) 64 (b) 78 (c) 50 (d) 57
28. 28.If a + b = 11 and ab = 28, what is a² + b²? (a) 121 (b) 177 (c) 93 (d) 65
29. 29.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 5625 (b) 7225 (c) 5635 (d) 6375
30. 30.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 1)(x + 8) (c) (x + 2)(x + 7) (d) (x + 1)(x + 14)
31. 31.What is the sum of the roots of x² − 14x + 48 = 0? (a) 15 (b) 48 (c) 13 (d) 14
32. 32.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x − 2)(x − 4) (c) (x + 2)(x + 4) (d) (x + 1)(x + 8)
33. 33.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x + 3)(x + 7) (c) (x + 1)(x + 9) (d) (x − 3)(x − 7)
34. 34.If a + b = 19 and ab = 84, what is (a − b)²? (a) 193 (b) 5 (c) 361 (d) 25
35. 35.What are the roots of x² − 13x + 42 = 0? (a) 2 and 21 (b) 1 and 42 (c) 1 and 12 (d) 6 and 7
36. 36.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 324 (b) 334 (c) 396 (d) 484
37. 37.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6889 (b) 6899 (c) 5929 (d) 6409
38. 38.What are the roots of x² − 12x + 32 = 0? (a) 2 and 16 (b) 1 and 32 (c) 1 and 11 (d) 4 and 8
39. 39.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3625 (b) 4225 (c) 3025 (d) 4235
40. 40.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1296 (b) 1306 (c) 1584 (d) 1936

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