Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 85 · 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 = 14 and ab = 45, what is (a − b)²? (a) 106 (b) 196 (c) 16 (d) 4
2. 2.If a + b = 8 and ab = 12, what is (a − b)²? (a) 16 (b) 4 (c) 64 (d) 40
3. 3.What is the product of the roots of x² − 10x + 21 = 0? (a) 10 (b) 20 (c) 21 (d) 22
4. 4.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 18) (b) (x + 1)(x + 10) (c) (x + 2)(x + 9) (d) (x − 2)(x − 9)
5. 5.If a + b = 6 and ab = 8, what is (a − b)²? (a) 4 (b) 36 (c) 2 (d) 20
6. 6.If a + b = 18 and ab = 80, what is (a − b)²? (a) 324 (b) 164 (c) 2 (d) 4
7. 7.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 7579 (b) 8649 (c) 7569 (d) 8091
8. 8.If a + b = 15 and ab = 54, what is (a − b)²? (a) 3 (b) 117 (c) 9 (d) 225
9. 9.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 1)(x + 5) (c) (x + 1)(x + 8) (d) (x + 2)(x + 4)
10. 10.If a + b = 18 and ab = 72, what is (a − b)²? (a) 6 (b) 180 (c) 324 (d) 36
11. 11.If a + b = 15 and ab = 56, what is (a − b)²? (a) 225 (b) 1 (c) 169 (d) 113
12. 12.What are the roots of x² − 10x + 24 = 0? (a) 1 and 9 (b) 1 and 24 (c) 4 and 6 (d) 2 and 12
13. 13.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6734 (b) 6724 (c) 6404 (d) 6084
14. 14.What is the product of the roots of x² − 13x + 42 = 0? (a) 43 (b) 13 (c) 41 (d) 42
15. 15.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 14 (c) 16 (d) 15
16. 16.If a + b = 7 and ab = 10, what is a² + b²? (a) 49 (b) 69 (c) 39 (d) 29
17. 17.If a + b = 7 and ab = 10, what is (a − b)²? (a) 29 (b) 49 (c) 9 (d) 3
18. 18.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2916 (b) 2126 (c) 2484 (d) 2116
19. 19.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x − 4)(x − 5) (c) (x + 1)(x + 8) (d) (x + 4)(x + 5)
20. 20.If a + b = 6 and ab = 8, what is a² + b²? (a) 20 (b) 52 (c) 36 (d) 28
21. 21.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 1 and 16 (c) 2 and 36 (d) 1 and 72
22. 22.What is the sum of the roots of x² − 13x + 42 = 0? (a) 12 (b) 14 (c) 13 (d) 42
23. 23.If a + b = 8 and ab = 15, what is (a − b)²? (a) 34 (b) 64 (c) 4 (d) 2
24. 24.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x − 5)(x − 6) (c) (x + 5)(x + 6) (d) (x + 1)(x + 10)
25. 25.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3491 (b) 3721 (c) 3599 (d) 3481
26. 26.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 4096 (b) 3146 (c) 3584 (d) 3136
27. 27.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 181 (c) 541 (d) 361
28. 28.If a + b = 16 and ab = 60, what is a² + b²? (a) 136 (b) 196 (c) 376 (d) 256
29. 29.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 3025 (b) 2025 (c) 2475 (d) 2035
30. 30.What is the product of the roots of x² − 13x + 40 = 0? (a) 41 (b) 40 (c) 39 (d) 13
31. 31.Which is the correct factorisation of x² + 13x + 40? (a) (x + 5)(x + 8) (b) (x − 5)(x − 8) (c) (x + 1)(x + 12) (d) (x + 1)(x + 40)
32. 32.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 1)(x + 13) (c) (x + 1)(x + 45) (d) (x + 5)(x + 9)
33. 33.If a + b = 11 and ab = 10, what is (a − b)²? (a) 101 (b) 9 (c) 121 (d) 81
34. 34.If a + b = 14 and ab = 40, what is (a − b)²? (a) 116 (b) 6 (c) 36 (d) 196
35. 35.If a + b = 11 and ab = 30, what is (a − b)²? (a) 91 (b) 121 (c) 61 (d) 1
36. 36.What are the roots of x² − 12x + 35 = 0? (a) 2 and 10 (b) 1 and 35 (c) 1 and 11 (d) 5 and 7
37. 37.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 739 (b) 1089 (c) 729 (d) 891
38. 38.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 334 (b) 396 (c) 484 (d) 324
39. 39.If a + b = 14 and ab = 48, what is (a − b)²? (a) 196 (b) 100 (c) 2 (d) 4
40. 40.What is the sum of the roots of x² − 12x + 35 = 0? (a) 11 (b) 35 (c) 13 (d) 12

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