Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 933 · 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 = 13 and ab = 22, what is (a − b)²? (a) 125 (b) 9 (c) 169 (d) 81
2. 2.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8091 (b) 7579 (c) 8649 (d) 7569
3. 3.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 841 (b) 851 (c) 961 (d) 899
4. 4.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 8084 (b) 8836 (c) 7406 (d) 7396
5. 5.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
6. 6.If a + b = 14 and ab = 24, what is a² + b²? (a) 148 (b) 244 (c) 172 (d) 196
7. 7.What are the roots of x² − 16x + 63 = 0? (a) 1 and 63 (b) 7 and 9 (c) 3 and 21 (d) 1 and 15
8. 8.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8291 (b) 8101 (c) 7921 (d) 8281
9. 9.If a + b = 18 and ab = 72, what is (a − b)²? (a) 324 (b) 6 (c) 36 (d) 180
10. 10.If a + b = 5 and ab = 6, what is a² + b²? (a) 25 (b) 13 (c) 37 (d) 19
11. 11.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 6384 (b) 5786 (c) 5776 (d) 7056
12. 12.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 1)(x + 9) (c) (x + 1)(x + 24) (d) (x + 4)(x + 6)
13. 13.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 18 (c) 72 (d) 17
14. 14.What is the sum of the roots of x² − 12x + 27 = 0? (a) 27 (b) 13 (c) 12 (d) 11
15. 15.If a + b = 12 and ab = 35, what is a² + b²? (a) 74 (b) 214 (c) 144 (d) 109
16. 16.If a + b = 16 and ab = 60, what is (a − b)²? (a) 256 (b) 136 (c) 4 (d) 16
17. 17.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)
18. 18.If a + b = 17 and ab = 66, what is (a − b)²? (a) 289 (b) 25 (c) 5 (d) 157
19. 19.If a + b = 8 and ab = 12, what is a² + b²? (a) 52 (b) 40 (c) 88 (d) 64
20. 20.If a + b = 16 and ab = 55, what is a² + b²? (a) 201 (b) 366 (c) 146 (d) 256
21. 21.If a + b = 12 and ab = 20, what is a² + b²? (a) 144 (b) 104 (c) 184 (d) 124
22. 22.What is the product of the roots of x² − 16x + 63 = 0? (a) 16 (b) 63 (c) 62 (d) 64
23. 23.Which is the correct factorisation of x² + 10x + 21? (a) (x − 3)(x − 7) (b) (x + 1)(x + 21) (c) (x + 1)(x + 9) (d) (x + 3)(x + 7)
24. 24.Which is the correct factorisation of x² + 9x + 14? (a) (x + 2)(x + 7) (b) (x + 1)(x + 14) (c) (x − 2)(x − 7) (d) (x + 1)(x + 8)
25. 25.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 299 (b) 391 (c) 529 (d) 289
26. 26.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2611 (b) 2501 (c) 2601 (d) 2401
27. 27.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 8125 (b) 9025 (c) 9035 (d) 7225
28. 28.If a + b = 9 and ab = 20, what is a² + b²? (a) 81 (b) 41 (c) 121 (d) 61
29. 29.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3601 (b) 3731 (c) 3481 (d) 3721
30. 30.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 361 (c) 1 (d) 181
31. 31.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 1)(x + 30) (c) (x + 1)(x + 10) (d) (x + 5)(x + 6)
32. 32.If a + b = 15 and ab = 36, what is (a − b)²? (a) 9 (b) 153 (c) 225 (d) 81
33. 33.If a + b = 5 and ab = 4, what is (a − b)²? (a) 3 (b) 25 (c) 9 (d) 17
34. 34.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 5625 (b) 6425 (c) 7235 (d) 7225
35. 35.If a + b = 6 and ab = 5, what is a² + b²? (a) 36 (b) 46 (c) 31 (d) 26
36. 36.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 7)(x + 8) (c) (x − 7)(x − 8) (d) (x + 1)(x + 56)
37. 37.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1764 (b) 1604 (c) 1774 (d) 1444
38. 38.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10826 (b) 9216 (c) 10816 (d) 10016
39. 39.What is the sum of the roots of x² − 11x + 24 = 0? (a) 11 (b) 24 (c) 10 (d) 12
40. 40.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1099 (b) 909 (c) 729 (d) 1089

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