Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 780 · 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 49²? (a) 2601 (b) 2401 (c) 2411 (d) 2499
2. 2.If a + b = 17 and ab = 70, what is a² + b²? (a) 219 (b) 289 (c) 149 (d) 429
3. 3.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3136 (b) 3584 (c) 4096 (d) 3146
4. 4.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x + 7)(x + 8) (d) (x − 7)(x − 8)
5. 5.If a + b = 12 and ab = 27, what is a² + b²? (a) 117 (b) 90 (c) 198 (d) 144
6. 6.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 71 (c) 73 (d) 72
7. 7.If a + b = 15 and ab = 36, what is a² + b²? (a) 297 (b) 153 (c) 189 (d) 225
8. 8.What is the sum of the roots of x² − 11x + 24 = 0? (a) 11 (b) 24 (c) 10 (d) 12
9. 9.If a + b = 5 and ab = 6, what is (a − b)²? (a) 13 (b) 19 (c) 1 (d) 25
10. 10.If a + b = 15 and ab = 44, what is (a − b)²? (a) 137 (b) 225 (c) 49 (d) 7
11. 11.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1946 (b) 1616 (c) 1936 (d) 1296
12. 12.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x + 5)(x + 9) (c) (x + 1)(x + 13) (d) (x − 5)(x − 9)
13. 13.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 5625 (b) 4225 (c) 4875 (d) 4235
14. 14.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 676 (b) 686 (c) 884 (d) 1156
15. 15.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6734 (b) 6404 (c) 6724 (d) 6084
16. 16.What are the roots of x² − 10x + 21 = 0? (a) 2 and 8 (b) 1 and 21 (c) 3 and 7 (d) 1 and 9
17. 17.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2035 (b) 1625 (c) 2025 (d) 1225
18. 18.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x + 1)(x + 7) (c) (x + 3)(x + 5) (d) (x − 3)(x − 5)
19. 19.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1369 (b) 1609 (c) 1849 (d) 1859
20. 20.What is the product of the roots of x² − 11x + 28 = 0? (a) 27 (b) 29 (c) 11 (d) 28
21. 21.If a + b = 8 and ab = 12, what is a² + b²? (a) 64 (b) 40 (c) 52 (d) 88
22. 22.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1369 (b) 1379 (c) 1849 (d) 1591
23. 23.If a + b = 9 and ab = 20, what is (a − b)²? (a) 1 (b) 81 (c) 41 (d) 61
24. 24.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3596 (b) 3844 (c) 3374 (d) 3364
25. 25.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
26. 26.If a + b = 12 and ab = 20, what is a² + b²? (a) 184 (b) 124 (c) 144 (d) 104
27. 27.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 10816 (b) 9984 (c) 9226 (d) 9216
28. 28.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 7)(x + 9) (c) (x − 7)(x − 9) (d) (x + 1)(x + 63)
29. 29.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 400 (c) 4 (d) 202
30. 30.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x + 1)(x + 21) (c) (x − 3)(x − 7) (d) (x + 3)(x + 7)
31. 31.What is the product of the roots of x² − 15x + 54 = 0? (a) 54 (b) 15 (c) 53 (d) 55
32. 32.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 635 (b) 875 (c) 625 (d) 1225
33. 33.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 5329 (b) 4499 (c) 4489 (d) 4891
34. 34.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 7225 (b) 9035 (c) 8125 (d) 9025
35. 35.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 7744 (b) 8104 (c) 8474 (d) 8464
36. 36.What are the roots of x² − 11x + 18 = 0? (a) 3 and 6 (b) 1 and 18 (c) 1 and 10 (d) 2 and 9
37. 37.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2401 (b) 2501 (c) 2601 (d) 2611
38. 38.If a + b = 9 and ab = 14, what is a² + b²? (a) 109 (b) 53 (c) 81 (d) 67
39. 39.If a + b = 13 and ab = 42, what is (a − b)²? (a) 127 (b) 85 (c) 169 (d) 1
40. 40.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x + 1)(x + 10) (c) (x − 4)(x − 7) (d) (x + 4)(x + 7)

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