Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 809 · 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 is the product of the roots of x² − 15x + 54 = 0? (a) 53 (b) 54 (c) 55 (d) 15
2. 2.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1225 (b) 925 (c) 1235 (d) 625
3. 3.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 1)(x + 18) (c) (x + 1)(x + 8) (d) (x + 3)(x + 6)
4. 4.If a + b = 20 and ab = 99, what is (a − b)²? (a) 4 (b) 400 (c) 202 (d) 2
5. 5.If a + b = 20 and ab = 99, what is a² + b²? (a) 400 (b) 301 (c) 202 (d) 598
6. 6.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8091 (b) 8649 (c) 7569 (d) 7579
7. 7.If a + b = 10 and ab = 16, what is (a − b)²? (a) 36 (b) 68 (c) 6 (d) 100
8. 8.If a + b = 16 and ab = 63, what is (a − b)²? (a) 2 (b) 256 (c) 4 (d) 130
9. 9.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 271 (c) 1 (d) 361
10. 10.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x − 4)(x − 5) (c) (x + 4)(x + 5) (d) (x + 1)(x + 8)
11. 11.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2601 (b) 2401 (c) 2501 (d) 2611
12. 12.If a + b = 12 and ab = 32, what is a² + b²? (a) 112 (b) 80 (c) 144 (d) 208
13. 13.If a + b = 12 and ab = 20, what is a² + b²? (a) 124 (b) 184 (c) 144 (d) 104
14. 14.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
15. 15.If a + b = 9 and ab = 20, what is a² + b²? (a) 41 (b) 61 (c) 121 (d) 81
16. 16.If a + b = 19 and ab = 88, what is (a − b)²? (a) 185 (b) 361 (c) 3 (d) 9
17. 17.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 16 (c) 63 (d) 62
18. 18.If a + b = 9 and ab = 14, what is a² + b²? (a) 67 (b) 53 (c) 81 (d) 109
19. 19.If a + b = 17 and ab = 60, what is (a − b)²? (a) 7 (b) 169 (c) 49 (d) 289
20. 20.What is the sum of the roots of x² − 10x + 24 = 0? (a) 24 (b) 9 (c) 11 (d) 10
21. 21.What are the roots of x² − 14x + 45 = 0? (a) 5 and 9 (b) 1 and 13 (c) 1 and 45 (d) 3 and 15
22. 22.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 2)(x + 7) (c) (x + 1)(x + 8) (d) (x + 1)(x + 14)
23. 23.What is the product of the roots of x² − 11x + 28 = 0? (a) 29 (b) 28 (c) 27 (d) 11
24. 24.If a + b = 3 and ab = 2, what is (a − b)²? (a) 5 (b) 7 (c) 1 (d) 9
25. 25.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)
26. 26.If a + b = 11 and ab = 10, what is a² + b²? (a) 121 (b) 111 (c) 141 (d) 101
27. 27.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 1)(x + 11) (c) (x + 1)(x + 35) (d) (x + 5)(x + 7)
28. 28.What is the product of the roots of x² − 14x + 48 = 0? (a) 49 (b) 48 (c) 47 (d) 14
29. 29.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 9025 (b) 8075 (c) 7235 (d) 7225
30. 30.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 10201 (b) 9811 (c) 9999 (d) 9801
31. 31.If a + b = 17 and ab = 72, what is (a − b)²? (a) 217 (b) 145 (c) 1 (d) 289
32. 32.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 63 (c) 15 (d) 17
33. 33.If a + b = 14 and ab = 24, what is (a − b)²? (a) 10 (b) 196 (c) 100 (d) 148
34. 34.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10404 (b) 10004 (c) 10414 (d) 9604
35. 35.If a + b = 11 and ab = 18, what is (a − b)²? (a) 7 (b) 85 (c) 121 (d) 49
36. 36.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 42) (c) (x + 6)(x + 7) (d) (x + 1)(x + 12)
37. 37.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x + 1)(x + 10) (c) (x + 1)(x + 30) (d) (x − 5)(x − 6)
38. 38.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 5476 (b) 4884 (c) 4366 (d) 4356
39. 39.If a + b = 13 and ab = 40, what is a² + b²? (a) 169 (b) 129 (c) 249 (d) 89
40. 40.If a + b = 11 and ab = 28, what is (a − b)²? (a) 121 (b) 65 (c) 3 (d) 9

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