Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 910 · 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 = 8 and ab = 7, what is (a − b)²? (a) 50 (b) 64 (c) 36 (d) 6
2. 2.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x − 4)(x − 9) (c) (x + 1)(x + 12) (d) (x + 4)(x + 9)
3. 3.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6899 (b) 6889 (c) 6409 (d) 5929
4. 4.If a + b = 17 and ab = 72, what is (a − b)²? (a) 217 (b) 1 (c) 145 (d) 289
5. 5.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9409 (b) 9991 (c) 9419 (d) 10609
6. 6.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x + 1)(x + 18) (c) (x − 2)(x − 9) (d) (x + 2)(x + 9)
7. 7.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x + 1)(x + 40) (c) (x + 5)(x + 8) (d) (x − 5)(x − 8)
8. 8.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6889 (b) 5929 (c) 5939 (d) 6391
9. 9.If a + b = 16 and ab = 63, what is (a − b)²? (a) 130 (b) 4 (c) 2 (d) 256
10. 10.If a + b = 11 and ab = 18, what is a² + b²? (a) 121 (b) 85 (c) 157 (d) 103
11. 11.If a + b = 18 and ab = 80, what is (a − b)²? (a) 164 (b) 4 (c) 324 (d) 2
12. 12.If a + b = 10 and ab = 21, what is (a − b)²? (a) 58 (b) 100 (c) 4 (d) 16
13. 13.What is the product of the roots of x² − 10x + 16 = 0? (a) 10 (b) 16 (c) 15 (d) 17
14. 14.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x + 1)(x + 13) (c) (x + 5)(x + 9) (d) (x − 5)(x − 9)
15. 15.If a + b = 11 and ab = 18, what is (a − b)²? (a) 7 (b) 85 (c) 49 (d) 121
16. 16.If a + b = 7 and ab = 10, what is (a − b)²? (a) 3 (b) 9 (c) 49 (d) 29
17. 17.If a + b = 7 and ab = 6, what is (a − b)²? (a) 37 (b) 49 (c) 5 (d) 25
18. 18.If a + b = 10 and ab = 9, what is a² + b²? (a) 118 (b) 100 (c) 82 (d) 91
19. 19.If a + b = 11 and ab = 24, what is (a − b)²? (a) 121 (b) 73 (c) 25 (d) 5
20. 20.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 400 (c) 301 (d) 202
21. 21.What is the sum of the roots of x² − 6x + 8 = 0? (a) 7 (b) 5 (c) 6 (d) 8
22. 22.What are the roots of x² − 12x + 27 = 0? (a) 2 and 10 (b) 3 and 9 (c) 1 and 11 (d) 1 and 27
23. 23.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 451 (b) 401 (c) 441 (d) 361
24. 24.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 5786 (b) 7056 (c) 5776 (d) 6384
25. 25.What are the roots of x² − 6x + 8 = 0? (a) 1 and 8 (b) 2 and 5 (c) 2 and 4 (d) 1 and 5
26. 26.What is the sum of the roots of x² − 14x + 48 = 0? (a) 15 (b) 14 (c) 13 (d) 48
27. 27.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 494 (b) 404 (c) 484 (d) 324
28. 28.If a + b = 16 and ab = 60, what is (a − b)²? (a) 4 (b) 256 (c) 136 (d) 16
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.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 1 and 14 (c) 2 and 28 (d) 7 and 8
31. 31.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 72) (c) (x + 1)(x + 16) (d) (x − 8)(x − 9)
32. 32.If a + b = 13 and ab = 42, what is a² + b²? (a) 169 (b) 85 (c) 253 (d) 127
33. 33.If a + b = 10 and ab = 9, what is (a − b)²? (a) 64 (b) 100 (c) 82 (d) 8
34. 34.If a + b = 6 and ab = 5, what is a² + b²? (a) 31 (b) 26 (c) 46 (d) 36
35. 35.If a + b = 4 and ab = 3, what is a² + b²? (a) 16 (b) 22 (c) 13 (d) 10
36. 36.If a + b = 15 and ab = 36, what is a² + b²? (a) 297 (b) 225 (c) 189 (d) 153
37. 37.What are the roots of x² − 11x + 24 = 0? (a) 2 and 12 (b) 3 and 8 (c) 1 and 24 (d) 1 and 10
38. 38.What are the roots of x² − 14x + 45 = 0? (a) 5 and 9 (b) 3 and 15 (c) 1 and 45 (d) 1 and 13
39. 39.What is the sum of the roots of x² − 9x + 18 = 0? (a) 18 (b) 8 (c) 9 (d) 10
40. 40.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 576 (b) 586 (c) 416 (d) 256

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