Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 927 · 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.What is the product of the roots of x² − 13x + 40 = 0? (a) 13 (b) 41 (c) 39 (d) 40
2. 2.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1609 (b) 1849 (c) 1859 (d) 1369
3. 3.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6241 (b) 6561 (c) 6251 (d) 6399
4. 4.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 271 (c) 181 (d) 361
5. 5.What are the roots of x² − 11x + 30 = 0? (a) 1 and 30 (b) 2 and 15 (c) 1 and 10 (d) 5 and 6
6. 6.If a + b = 11 and ab = 28, what is (a − b)²? (a) 3 (b) 65 (c) 121 (d) 9
7. 7.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 5929 (b) 6409 (c) 6899 (d) 6889
8. 8.If a + b = 13 and ab = 12, what is a² + b²? (a) 169 (b) 145 (c) 157 (d) 193
9. 9.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x + 5)(x + 9) (c) (x − 5)(x − 9) (d) (x + 1)(x + 13)
10. 10.What are the roots of x² − 7x + 12 = 0? (a) 1 and 12 (b) 2 and 6 (c) 1 and 6 (d) 3 and 4
11. 11.What is the product of the roots of x² − 11x + 28 = 0? (a) 29 (b) 27 (c) 11 (d) 28
12. 12.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 904 (b) 1034 (c) 1024 (d) 784
13. 13.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 7396 (b) 8836 (c) 8084 (d) 7406
14. 14.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 324 (b) 334 (c) 484 (d) 396
15. 15.What are the roots of x² − 9x + 14 = 0? (a) 1 and 8 (b) 3 and 6 (c) 1 and 14 (d) 2 and 7
16. 16.If a + b = 17 and ab = 72, what is a² + b²? (a) 217 (b) 289 (c) 433 (d) 145
17. 17.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4904 (b) 4624 (c) 5194 (d) 5184
18. 18.If a + b = 13 and ab = 22, what is a² + b²? (a) 147 (b) 169 (c) 213 (d) 125
19. 19.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3146 (b) 3136 (c) 3584 (d) 4096
20. 20.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x − 3)(x − 7) (c) (x + 1)(x + 21) (d) (x + 3)(x + 7)
21. 21.What is the product of the roots of x² − 7x + 12 = 0? (a) 11 (b) 12 (c) 13 (d) 7
22. 22.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2504 (b) 2704 (c) 2714 (d) 2304
23. 23.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 5476 (b) 5486 (c) 4916 (d) 4356
24. 24.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x + 1)(x + 6) (c) (x − 2)(x − 3) (d) (x + 2)(x + 3)
25. 25.Which is the correct factorisation of x² + 12x + 27? (a) (x + 3)(x + 9) (b) (x + 1)(x + 11) (c) (x − 3)(x − 9) (d) (x + 1)(x + 27)
26. 26.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1625 (b) 1225 (c) 2025 (d) 2035
27. 27.If a + b = 20 and ab = 99, what is a² + b²? (a) 202 (b) 400 (c) 598 (d) 301
28. 28.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8109 (b) 8659 (c) 7569 (d) 8649
29. 29.What is the sum of the roots of x² − 10x + 16 = 0? (a) 9 (b) 10 (c) 11 (d) 16
30. 30.If a + b = 16 and ab = 63, what is (a − b)²? (a) 256 (b) 2 (c) 130 (d) 4
31. 31.If a + b = 17 and ab = 66, what is a² + b²? (a) 157 (b) 289 (c) 223 (d) 421
32. 32.If a + b = 3 and ab = 2, what is a² + b²? (a) 5 (b) 9 (c) 13 (d) 7
33. 33.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9226 (b) 9984 (c) 10816 (d) 9216
34. 34.What are the roots of x² − 13x + 40 = 0? (a) 2 and 20 (b) 1 and 40 (c) 1 and 12 (d) 5 and 8
35. 35.What is the sum of the roots of x² − 12x + 32 = 0? (a) 12 (b) 32 (c) 11 (d) 13
36. 36.What is the sum of the roots of x² − 10x + 24 = 0? (a) 9 (b) 24 (c) 10 (d) 11
37. 37.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 1156 (b) 686 (c) 676 (d) 884
38. 38.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 441 (b) 399 (c) 361 (d) 371
39. 39.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 971 (b) 961 (c) 901 (d) 841
40. 40.What are the roots of x² − 13x + 42 = 0? (a) 1 and 42 (b) 1 and 12 (c) 2 and 21 (d) 6 and 7

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