Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 599 · 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.Which is the correct factorisation of x² + 10x + 21? (a) (x + 3)(x + 7) (b) (x + 1)(x + 9) (c) (x − 3)(x − 7) (d) (x + 1)(x + 21)
2. 2.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)
3. 3.If a + b = 5 and ab = 4, what is a² + b²? (a) 25 (b) 17 (c) 33 (d) 21
4. 4.Which is the correct factorisation of x² + 7x + 10? (a) (x + 2)(x + 5) (b) (x + 1)(x + 10) (c) (x − 2)(x − 5) (d) (x + 1)(x + 6)
5. 5.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1521 (b) 1691 (c) 1601 (d) 1681
6. 6.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3146 (b) 3584 (c) 3136 (d) 4096
7. 7.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 2 and 36 (c) 1 and 72 (d) 8 and 9
8. 8.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6391 (b) 5929 (c) 6889 (d) 5939
9. 9.If a + b = 16 and ab = 48, what is a² + b²? (a) 256 (b) 352 (c) 208 (d) 160
10. 10.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x − 7)(x − 9) (c) (x + 1)(x + 63) (d) (x + 1)(x + 15)
11. 11.If a + b = 13 and ab = 42, what is (a − b)²? (a) 127 (b) 85 (c) 169 (d) 1
12. 12.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2504 (b) 2304 (c) 2714 (d) 2704
13. 13.If a + b = 19 and ab = 88, what is a² + b²? (a) 273 (b) 185 (c) 361 (d) 537
14. 14.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x − 2)(x − 9) (c) (x + 2)(x + 9) (d) (x + 1)(x + 18)
15. 15.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 1)(x + 42) (c) (x − 6)(x − 7) (d) (x + 6)(x + 7)
16. 16.If a + b = 14 and ab = 45, what is (a − b)²? (a) 106 (b) 196 (c) 4 (d) 16
17. 17.What is the product of the roots of x² − 11x + 18 = 0? (a) 11 (b) 17 (c) 19 (d) 18
18. 18.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8474 (b) 8104 (c) 8464 (d) 7744
19. 19.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10404 (b) 10414 (c) 9604 (d) 10004
20. 20.What are the roots of x² − 15x + 54 = 0? (a) 1 and 14 (b) 1 and 54 (c) 6 and 9 (d) 2 and 27
21. 21.If a + b = 15 and ab = 50, what is a² + b²? (a) 125 (b) 225 (c) 325 (d) 175
22. 22.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 6)(x + 8) (c) (x + 1)(x + 13) (d) (x + 1)(x + 48)
23. 23.If a + b = 19 and ab = 84, what is (a − b)²? (a) 361 (b) 5 (c) 25 (d) 193
24. 24.If a + b = 14 and ab = 33, what is (a − b)²? (a) 64 (b) 8 (c) 130 (d) 196
25. 25.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2035 (b) 1625 (c) 2025 (d) 1225
26. 26.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7056 (b) 6416 (c) 5776 (d) 7066
27. 27.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 16 (c) 15 (d) 14
28. 28.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 404 (b) 494 (c) 484 (d) 324
29. 29.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)
30. 30.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 884 (b) 1156 (c) 676 (d) 686
31. 31.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 729 (b) 739 (c) 1089 (d) 891
32. 32.What is the sum of the roots of x² − 13x + 42 = 0? (a) 12 (b) 14 (c) 42 (d) 13
33. 33.If a + b = 18 and ab = 80, what is (a − b)²? (a) 4 (b) 164 (c) 324 (d) 2
34. 34.If a + b = 16 and ab = 63, what is (a − b)²? (a) 256 (b) 130 (c) 4 (d) 2
35. 35.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 416 (b) 586 (c) 256 (d) 576
36. 36.If a + b = 9 and ab = 18, what is a² + b²? (a) 117 (b) 63 (c) 45 (d) 81
37. 37.If a + b = 17 and ab = 60, what is (a − b)²? (a) 7 (b) 289 (c) 49 (d) 169
38. 38.What is the sum of the roots of x² − 11x + 18 = 0? (a) 11 (b) 12 (c) 10 (d) 18
39. 39.If a + b = 15 and ab = 44, what is (a − b)²? (a) 49 (b) 225 (c) 137 (d) 7
40. 40.If a + b = 11 and ab = 30, what is a² + b²? (a) 181 (b) 121 (c) 91 (d) 61

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