Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 892 · 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 + 13x + 36? (a) (x + 1)(x + 36) (b) (x + 1)(x + 12) (c) (x − 4)(x − 9) (d) (x + 4)(x + 9)
2. 2.If a + b = 12 and ab = 27, what is a² + b²? (a) 198 (b) 117 (c) 90 (d) 144
3. 3.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 63 (c) 15 (d) 17
4. 4.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 925 (b) 1235 (c) 625 (d) 1225
5. 5.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 7569 (b) 8659 (c) 8649 (d) 8109
6. 6.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 1 and 16 (c) 2 and 36 (d) 1 and 72
7. 7.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)
8. 8.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3025 (b) 3625 (c) 4225 (d) 4235
9. 9.If a + b = 9 and ab = 14, what is (a − b)²? (a) 5 (b) 81 (c) 25 (d) 53
10. 10.If a + b = 18 and ab = 77, what is a² + b²? (a) 324 (b) 247 (c) 170 (d) 478
11. 11.If a + b = 10 and ab = 21, what is a² + b²? (a) 142 (b) 58 (c) 79 (d) 100
12. 12.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1591 (b) 1849 (c) 1369 (d) 1379
13. 13.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 73 (c) 71 (d) 72
14. 14.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1764 (b) 1444 (c) 1604 (d) 1774
15. 15.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 3025 (b) 2525 (c) 2025 (d) 3035
16. 16.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)
17. 17.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2809 (b) 2219 (c) 2491 (d) 2209
18. 18.What are the roots of x² − 5x + 6 = 0? (a) 2 and 4 (b) 1 and 6 (c) 1 and 4 (d) 2 and 3
19. 19.Which is the correct factorisation of x² + 8x + 12? (a) (x + 2)(x + 6) (b) (x + 1)(x + 7) (c) (x + 1)(x + 12) (d) (x − 2)(x − 6)
20. 20.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 7579 (b) 7569 (c) 8091 (d) 8649
21. 21.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 9) (b) (x − 4)(x − 6) (c) (x + 4)(x + 6) (d) (x + 1)(x + 24)
22. 22.What is the sum of the roots of x² − 14x + 48 = 0? (a) 14 (b) 13 (c) 48 (d) 15
23. 23.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4925 (b) 4225 (c) 5635 (d) 5625
24. 24.If a + b = 8 and ab = 15, what is a² + b²? (a) 94 (b) 64 (c) 49 (d) 34
25. 25.If a + b = 17 and ab = 66, what is (a − b)²? (a) 25 (b) 157 (c) 5 (d) 289
26. 26.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 901 (b) 971 (c) 841 (d) 961
27. 27.What are the roots of x² − 12x + 32 = 0? (a) 2 and 16 (b) 1 and 32 (c) 1 and 11 (d) 4 and 8
28. 28.What is the product of the roots of x² − 15x + 56 = 0? (a) 56 (b) 57 (c) 55 (d) 15
29. 29.If a + b = 8 and ab = 7, what is (a − b)²? (a) 36 (b) 6 (c) 50 (d) 64
30. 30.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 896 (c) 1024 (d) 784
31. 31.What is the product of the roots of x² − 9x + 14 = 0? (a) 13 (b) 14 (c) 15 (d) 9
32. 32.If a + b = 4 and ab = 3, what is (a − b)²? (a) 16 (b) 10 (c) 4 (d) 2
33. 33.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 1)(x + 63) (c) (x + 7)(x + 9) (d) (x − 7)(x − 9)
34. 34.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
35. 35.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8096 (b) 8464 (c) 7754 (d) 7744
36. 36.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x − 3)(x − 7) (c) (x + 3)(x + 7) (d) (x + 1)(x + 21)
37. 37.If a + b = 16 and ab = 55, what is (a − b)²? (a) 146 (b) 6 (c) 36 (d) 256
38. 38.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2916 (b) 2484 (c) 2116 (d) 2126
39. 39.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 625 (b) 235 (c) 225 (d) 375
40. 40.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 9025 (b) 8075 (c) 7235 (d) 7225

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