Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 194 · 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 = 19 and ab = 84, what is (a − b)²? (a) 25 (b) 5 (c) 361 (d) 193
2. 2.What is the sum of the roots of x² − 7x + 12 = 0? (a) 8 (b) 12 (c) 7 (d) 6
3. 3.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 494 (b) 404 (c) 484 (d) 324
4. 4.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1235 (b) 625 (c) 925 (d) 1225
5. 5.What are the roots of x² − 11x + 30 = 0? (a) 1 and 10 (b) 1 and 30 (c) 2 and 15 (d) 5 and 6
6. 6.What is the sum of the roots of x² − 13x + 40 = 0? (a) 12 (b) 14 (c) 40 (d) 13
7. 7.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 7396 (b) 8116 (c) 8836 (d) 8846
8. 8.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 1)(x + 12) (c) (x + 6)(x + 7) (d) (x − 6)(x − 7)
9. 9.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 1 and 72 (c) 2 and 36 (d) 8 and 9
10. 10.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 5635 (b) 7225 (c) 6375 (d) 5625
11. 11.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6391 (b) 5929 (c) 6889 (d) 5939
12. 12.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 529 (b) 289 (c) 539 (d) 409
13. 13.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)
14. 14.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2314 (b) 2704 (c) 2304 (d) 2496
15. 15.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3136 (b) 3146 (c) 4096 (d) 3584
16. 16.What is the product of the roots of x² − 6x + 8 = 0? (a) 7 (b) 6 (c) 9 (d) 8
17. 17.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6734 (b) 6724 (c) 6404 (d) 6084
18. 18.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x + 1)(x + 16)
19. 19.If a + b = 13 and ab = 30, what is a² + b²? (a) 139 (b) 169 (c) 109 (d) 229
20. 20.If a + b = 12 and ab = 32, what is a² + b²? (a) 112 (b) 80 (c) 208 (d) 144
21. 21.If a + b = 6 and ab = 5, what is a² + b²? (a) 26 (b) 46 (c) 31 (d) 36
22. 22.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8104 (b) 8474 (c) 7744 (d) 8464
23. 23.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 7406 (b) 7396 (c) 8084 (d) 8836
24. 24.Which is the correct factorisation of x² + 11x + 24? (a) (x − 3)(x − 8) (b) (x + 1)(x + 10) (c) (x + 3)(x + 8) (d) (x + 1)(x + 24)
25. 25.If a + b = 5 and ab = 6, what is a² + b²? (a) 19 (b) 13 (c) 37 (d) 25
26. 26.If a + b = 7 and ab = 10, what is a² + b²? (a) 69 (b) 29 (c) 49 (d) 39
27. 27.If a + b = 15 and ab = 54, what is (a − b)²? (a) 117 (b) 3 (c) 225 (d) 9
28. 28.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2116 (b) 2916 (c) 2926 (d) 2516
29. 29.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7225 (b) 9025 (c) 8075 (d) 7235
30. 30.What are the roots of x² − 7x + 10 = 0? (a) 1 and 10 (b) 1 and 6 (c) 3 and 4 (d) 2 and 5
31. 31.If a + b = 16 and ab = 60, what is (a − b)²? (a) 256 (b) 136 (c) 4 (d) 16
32. 32.What is the sum of the roots of x² − 12x + 35 = 0? (a) 11 (b) 12 (c) 13 (d) 35
33. 33.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4899 (b) 5041 (c) 4771 (d) 4761
34. 34.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1521 (b) 1599 (c) 1681 (d) 1531
35. 35.If a + b = 12 and ab = 20, what is (a − b)²? (a) 8 (b) 64 (c) 104 (d) 144
36. 36.What is the product of the roots of x² − 8x + 12 = 0? (a) 8 (b) 12 (c) 11 (d) 13
37. 37.What is the sum of the roots of x² − 11x + 28 = 0? (a) 12 (b) 28 (c) 10 (d) 11
38. 38.Which is the correct factorisation of x² + 6x + 8? (a) (x + 2)(x + 4) (b) (x + 1)(x + 8) (c) (x − 2)(x − 4) (d) (x + 1)(x + 5)
39. 39.If a + b = 10 and ab = 24, what is (a − b)²? (a) 52 (b) 4 (c) 2 (d) 100
40. 40.If a + b = 17 and ab = 60, what is a² + b²? (a) 169 (b) 229 (c) 289 (d) 409

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