Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 796 · 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) 2525 (b) 3025 (c) 3035 (d) 2025
2. 2.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1521 (b) 1681 (c) 1599 (d) 1531
3. 3.If a + b = 11 and ab = 28, what is a² + b²? (a) 65 (b) 93 (c) 121 (d) 177
4. 4.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x − 7)(x − 8) (c) (x + 1)(x + 56) (d) (x + 1)(x + 14)
5. 5.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2116 (b) 2916 (c) 2126 (d) 2484
6. 6.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 6)(x + 8) (c) (x − 6)(x − 8) (d) (x + 1)(x + 48)
7. 7.What are the roots of x² − 11x + 30 = 0? (a) 1 and 10 (b) 1 and 30 (c) 5 and 6 (d) 2 and 15
8. 8.If a + b = 8 and ab = 7, what is (a − b)²? (a) 36 (b) 50 (c) 64 (d) 6
9. 9.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1691 (b) 1521 (c) 1601 (d) 1681
10. 10.If a + b = 13 and ab = 42, what is a² + b²? (a) 253 (b) 127 (c) 169 (d) 85
11. 11.If a + b = 18 and ab = 80, what is (a − b)²? (a) 324 (b) 164 (c) 2 (d) 4
12. 12.Which is the correct factorisation of x² + 9x + 18? (a) (x + 3)(x + 6) (b) (x − 3)(x − 6) (c) (x + 1)(x + 8) (d) (x + 1)(x + 18)
13. 13.If a + b = 6 and ab = 5, what is (a − b)²? (a) 36 (b) 26 (c) 4 (d) 16
14. 14.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3249 (b) 3591 (c) 3969 (d) 3259
15. 15.If a + b = 11 and ab = 28, what is (a − b)²? (a) 9 (b) 65 (c) 3 (d) 121
16. 16.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 56 (c) 57 (d) 15
17. 17.What are the roots of x² − 13x + 42 = 0? (a) 2 and 21 (b) 1 and 12 (c) 6 and 7 (d) 1 and 42
18. 18.If a + b = 17 and ab = 70, what is a² + b²? (a) 149 (b) 219 (c) 289 (d) 429
19. 19.What is the product of the roots of x² − 8x + 12 = 0? (a) 8 (b) 11 (c) 13 (d) 12
20. 20.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x − 5)(x − 9) (c) (x + 1)(x + 45) (d) (x + 5)(x + 9)
21. 21.If a + b = 5 and ab = 6, what is (a − b)²? (a) 25 (b) 13 (c) 1 (d) 19
22. 22.What are the roots of x² − 16x + 63 = 0? (a) 1 and 15 (b) 7 and 9 (c) 1 and 63 (d) 3 and 21
23. 23.If a + b = 17 and ab = 72, what is a² + b²? (a) 289 (b) 433 (c) 217 (d) 145
24. 24.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x + 1)(x + 24) (c) (x + 3)(x + 8) (d) (x − 3)(x − 8)
25. 25.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 1)(x + 11) (c) (x + 5)(x + 7) (d) (x + 1)(x + 35)
26. 26.What is the sum of the roots of x² − 8x + 15 = 0? (a) 8 (b) 9 (c) 15 (d) 7
27. 27.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 384 (b) 266 (c) 256 (d) 576
28. 28.If a + b = 6 and ab = 8, what is a² + b²? (a) 20 (b) 28 (c) 36 (d) 52
29. 29.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3575 (b) 3035 (c) 3025 (d) 4225
30. 30.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2611 (b) 2501 (c) 2601 (d) 2401
31. 31.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 5184 (b) 5194 (c) 4904 (d) 4624
32. 32.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 1 and 72 (c) 1 and 16 (d) 2 and 36
33. 33.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x + 3)(x + 7) (c) (x + 1)(x + 21) (d) (x − 3)(x − 7)
34. 34.If a + b = 10 and ab = 9, what is a² + b²? (a) 91 (b) 100 (c) 118 (d) 82
35. 35.If a + b = 7 and ab = 12, what is (a − b)²? (a) 49 (b) 1 (c) 25 (d) 37
36. 36.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x + 1)(x + 8) (c) (x − 2)(x − 7) (d) (x + 2)(x + 7)
37. 37.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 729 (b) 1089 (c) 891 (d) 739
38. 38.If a + b = 15 and ab = 56, what is (a − b)²? (a) 113 (b) 169 (c) 1 (d) 225
39. 39.What is the sum of the roots of x² − 13x + 36 = 0? (a) 12 (b) 13 (c) 36 (d) 14
40. 40.If a + b = 12 and ab = 27, what is a² + b²? (a) 117 (b) 198 (c) 90 (d) 144

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