Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 392 · 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 = 17 and ab = 70, what is (a − b)²? (a) 3 (b) 149 (c) 9 (d) 289
2. 2.If a + b = 6 and ab = 8, what is a² + b²? (a) 36 (b) 20 (c) 52 (d) 28
3. 3.What is the sum of the roots of x² − 9x + 18 = 0? (a) 8 (b) 9 (c) 10 (d) 18
4. 4.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x − 6)(x − 8) (c) (x + 6)(x + 8) (d) (x + 1)(x + 13)
5. 5.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4356 (b) 4366 (c) 5476 (d) 4884
6. 6.If a + b = 13 and ab = 40, what is (a − b)²? (a) 89 (b) 169 (c) 3 (d) 9
7. 7.If a + b = 3 and ab = 2, what is (a − b)²? (a) 1 (b) 5 (c) 7 (d) 9
8. 8.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 16)
9. 9.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2304 (b) 2504 (c) 2704 (d) 2714
10. 10.What is the product of the roots of x² − 13x + 42 = 0? (a) 43 (b) 41 (c) 42 (d) 13
11. 11.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 441 (c) 9 (d) 3
12. 12.What is the sum of the roots of x² − 14x + 48 = 0? (a) 15 (b) 13 (c) 14 (d) 48
13. 13.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3025 (b) 3035 (c) 3575 (d) 4225
14. 14.What are the roots of x² − 15x + 56 = 0? (a) 7 and 8 (b) 1 and 56 (c) 1 and 14 (d) 2 and 28
15. 15.If a + b = 16 and ab = 63, what is a² + b²? (a) 256 (b) 130 (c) 193 (d) 382
16. 16.If a + b = 15 and ab = 56, what is a² + b²? (a) 169 (b) 113 (c) 225 (d) 337
17. 17.If a + b = 12 and ab = 11, what is a² + b²? (a) 122 (b) 144 (c) 133 (d) 166
18. 18.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10609 (b) 10009 (c) 10619 (d) 9409
19. 19.If a + b = 20 and ab = 96, what is a² + b²? (a) 592 (b) 208 (c) 304 (d) 400
20. 20.What is the sum of the roots of x² − 13x + 42 = 0? (a) 42 (b) 13 (c) 12 (d) 14
21. 21.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 784 (b) 1034 (c) 904 (d) 1024
22. 22.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 7921 (b) 7931 (c) 8281 (d) 8099
23. 23.If a + b = 14 and ab = 45, what is a² + b²? (a) 151 (b) 106 (c) 286 (d) 196
24. 24.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x + 1)(x + 8) (c) (x + 2)(x + 4) (d) (x − 2)(x − 4)
25. 25.Which is the correct factorisation of x² + 8x + 15? (a) (x + 3)(x + 5) (b) (x + 1)(x + 15) (c) (x − 3)(x − 5) (d) (x + 1)(x + 7)
26. 26.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 5)(x + 6) (c) (x + 1)(x + 10) (d) (x − 5)(x − 6)
27. 27.What is the sum of the roots of x² − 11x + 30 = 0? (a) 30 (b) 12 (c) 11 (d) 10
28. 28.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 324 (b) 494 (c) 484 (d) 404
29. 29.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 8 and 9 (c) 1 and 16 (d) 1 and 72
30. 30.If a + b = 6 and ab = 5, what is a² + b²? (a) 36 (b) 46 (c) 31 (d) 26
31. 31.If a + b = 4 and ab = 3, what is (a − b)²? (a) 2 (b) 4 (c) 16 (d) 10
32. 32.Which is the correct factorisation of x² + 9x + 18? (a) (x + 3)(x + 6) (b) (x − 3)(x − 6) (c) (x + 1)(x + 18) (d) (x + 1)(x + 8)
33. 33.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6899 (b) 6889 (c) 5929 (d) 6409
34. 34.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x − 7)(x − 9) (c) (x + 1)(x + 15) (d) (x + 1)(x + 63)
35. 35.What are the roots of x² − 10x + 21 = 0? (a) 1 and 21 (b) 1 and 9 (c) 3 and 7 (d) 2 and 8
36. 36.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)
37. 37.If a + b = 9 and ab = 18, what is a² + b²? (a) 117 (b) 81 (c) 63 (d) 45
38. 38.If a + b = 14 and ab = 45, what is (a − b)²? (a) 4 (b) 106 (c) 16 (d) 196
39. 39.If a + b = 10 and ab = 16, what is (a − b)²? (a) 6 (b) 68 (c) 100 (d) 36
40. 40.If a + b = 7 and ab = 12, what is a² + b²? (a) 49 (b) 25 (c) 73 (d) 37

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