Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 342 · 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 34²? (a) 1166 (b) 676 (c) 1156 (d) 916
2. 2.If a + b = 10 and ab = 21, what is a² + b²? (a) 79 (b) 58 (c) 142 (d) 100
3. 3.What is the product of the roots of x² − 7x + 10 = 0? (a) 11 (b) 10 (c) 9 (d) 7
4. 4.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8836 (b) 8116 (c) 8846 (d) 7396
5. 5.What is the sum of the roots of x² − 14x + 45 = 0? (a) 15 (b) 45 (c) 14 (d) 13
6. 6.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 7406 (b) 8836 (c) 7396 (d) 8084
7. 7.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 16 (d) 17
8. 8.If a + b = 12 and ab = 20, what is (a − b)²? (a) 104 (b) 144 (c) 64 (d) 8
9. 9.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 17 (c) 72 (d) 71
10. 10.If a + b = 12 and ab = 32, what is (a − b)²? (a) 80 (b) 4 (c) 144 (d) 16
11. 11.What are the roots of x² − 8x + 12 = 0? (a) 3 and 4 (b) 2 and 6 (c) 1 and 7 (d) 1 and 12
12. 12.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 5329 (b) 4909 (c) 4489 (d) 5339
13. 13.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2484 (b) 2916 (c) 2126 (d) 2116
14. 14.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1444 (b) 1764 (c) 1596 (d) 1454
15. 15.If a + b = 16 and ab = 55, what is a² + b²? (a) 256 (b) 366 (c) 201 (d) 146
16. 16.If a + b = 7 and ab = 12, what is (a − b)²? (a) 1 (b) 25 (c) 49 (d) 37
17. 17.If a + b = 17 and ab = 70, what is (a − b)²? (a) 289 (b) 3 (c) 9 (d) 149
18. 18.If a + b = 12 and ab = 11, what is a² + b²? (a) 144 (b) 122 (c) 133 (d) 166
19. 19.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 1024 (c) 896 (d) 784
20. 20.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9984 (b) 9226 (c) 10816 (d) 9216
21. 21.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2304 (b) 2496 (c) 2704 (d) 2314
22. 22.If a + b = 8 and ab = 12, what is (a − b)²? (a) 40 (b) 64 (c) 16 (d) 4
23. 23.What are the roots of x² − 14x + 45 = 0? (a) 1 and 13 (b) 1 and 45 (c) 5 and 9 (d) 3 and 15
24. 24.What are the roots of x² − 6x + 8 = 0? (a) 1 and 5 (b) 2 and 5 (c) 2 and 4 (d) 1 and 8
25. 25.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 14) (c) (x + 1)(x + 56) (d) (x + 7)(x + 8)
26. 26.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 63) (c) (x + 7)(x + 9) (d) (x + 1)(x + 15)
27. 27.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 1)(x + 20) (c) (x + 1)(x + 8) (d) (x + 4)(x + 5)
28. 28.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 9035 (b) 9025 (c) 7225 (d) 8125
29. 29.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2704 (b) 2304 (c) 2714 (d) 2504
30. 30.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 56 (c) 15 (d) 57
31. 31.What are the roots of x² − 11x + 30 = 0? (a) 5 and 6 (b) 1 and 30 (c) 2 and 15 (d) 1 and 10
32. 32.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1625 (b) 1225 (c) 2025 (d) 2035
33. 33.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x − 3)(x − 7) (c) (x + 1)(x + 9) (d) (x + 3)(x + 7)
34. 34.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x + 1)(x + 27) (c) (x − 3)(x − 9) (d) (x + 3)(x + 9)
35. 35.If a + b = 17 and ab = 66, what is (a − b)²? (a) 25 (b) 157 (c) 289 (d) 5
36. 36.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3364 (b) 3844 (c) 3854 (d) 3604
37. 37.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 484 (b) 404 (c) 324 (d) 494
38. 38.If a + b = 13 and ab = 42, what is (a − b)²? (a) 169 (b) 85 (c) 1 (d) 127
39. 39.If a + b = 19 and ab = 88, what is a² + b²? (a) 273 (b) 361 (c) 185 (d) 537
40. 40.What is the product of the roots of x² − 9x + 14 = 0? (a) 9 (b) 13 (c) 15 (d) 14

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