Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 22 · 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 62²? (a) 3364 (b) 3604 (c) 3854 (d) 3844
2. 2.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 334 (b) 484 (c) 396 (d) 324
3. 3.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 2025 (b) 1225 (c) 1235 (d) 1575
4. 4.If a + b = 21 and ab = 108, what is a² + b²? (a) 657 (b) 225 (c) 441 (d) 333
5. 5.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x − 4)(x − 5) (c) (x + 4)(x + 5) (d) (x + 1)(x + 8)
6. 6.If a + b = 13 and ab = 12, what is (a − b)²? (a) 11 (b) 121 (c) 169 (d) 145
7. 7.If a + b = 12 and ab = 35, what is (a − b)²? (a) 2 (b) 4 (c) 144 (d) 74
8. 8.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 1 (c) 271 (d) 361
9. 9.If a + b = 15 and ab = 54, what is a² + b²? (a) 225 (b) 117 (c) 171 (d) 333
10. 10.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 235 (b) 625 (c) 225 (d) 375
11. 11.If a + b = 18 and ab = 80, what is a² + b²? (a) 244 (b) 324 (c) 484 (d) 164
12. 12.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x − 3)(x − 8) (c) (x + 3)(x + 8) (d) (x + 1)(x + 24)
13. 13.If a + b = 16 and ab = 63, what is (a − b)²? (a) 2 (b) 4 (c) 130 (d) 256
14. 14.What is the product of the roots of x² − 12x + 32 = 0? (a) 33 (b) 12 (c) 31 (d) 32
15. 15.If a + b = 9 and ab = 18, what is (a − b)²? (a) 81 (b) 9 (c) 45 (d) 3
16. 16.What is the sum of the roots of x² − 9x + 14 = 0? (a) 14 (b) 10 (c) 9 (d) 8
17. 17.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x − 6)(x − 9) (c) (x + 1)(x + 54) (d) (x + 6)(x + 9)
18. 18.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 441 (b) 371 (c) 399 (d) 361
19. 19.If a + b = 14 and ab = 33, what is a² + b²? (a) 196 (b) 163 (c) 130 (d) 262
20. 20.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 7225 (b) 5635 (c) 6375 (d) 5625
21. 21.What are the roots of x² − 15x + 54 = 0? (a) 1 and 54 (b) 1 and 14 (c) 2 and 27 (d) 6 and 9
22. 22.What is the sum of the roots of x² − 9x + 20 = 0? (a) 9 (b) 20 (c) 10 (d) 8
23. 23.What is the sum of the roots of x² − 12x + 35 = 0? (a) 11 (b) 35 (c) 12 (d) 13
24. 24.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2809 (b) 2509 (c) 2819 (d) 2209
25. 25.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 6)(x + 7) (c) (x + 1)(x + 42) (d) (x + 1)(x + 12)
26. 26.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8281 (b) 8099 (c) 7931 (d) 7921
27. 27.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x − 5)(x − 8) (c) (x + 1)(x + 40) (d) (x + 5)(x + 8)
28. 28.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x + 1)(x + 15) (c) (x − 3)(x − 5) (d) (x + 3)(x + 5)
29. 29.What is the sum of the roots of x² − 12x + 27 = 0? (a) 12 (b) 11 (c) 13 (d) 27
30. 30.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 1156 (b) 676 (c) 686 (d) 884
31. 31.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 11) (b) (x − 4)(x − 8) (c) (x + 4)(x + 8) (d) (x + 1)(x + 32)
32. 32.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4624 (b) 4904 (c) 5184 (d) 5194
33. 33.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2401 (b) 2611 (c) 2501 (d) 2601
34. 34.What is the product of the roots of x² − 9x + 14 = 0? (a) 15 (b) 13 (c) 9 (d) 14
35. 35.If a + b = 14 and ab = 24, what is (a − b)²? (a) 10 (b) 100 (c) 148 (d) 196
36. 36.If a + b = 9 and ab = 8, what is (a − b)²? (a) 81 (b) 49 (c) 65 (d) 7
37. 37.If a + b = 16 and ab = 60, what is (a − b)²? (a) 4 (b) 136 (c) 256 (d) 16
38. 38.What is the sum of the roots of x² − 11x + 30 = 0? (a) 10 (b) 11 (c) 12 (d) 30
39. 39.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 324 (b) 404 (c) 484 (d) 494
40. 40.What are the roots of x² − 8x + 12 = 0? (a) 2 and 6 (b) 1 and 12 (c) 3 and 4 (d) 1 and 7

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