Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 453 · 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 = 9 and ab = 8, what is a² + b²? (a) 65 (b) 73 (c) 81 (d) 97
2. 2.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2496 (b) 2314 (c) 2304 (d) 2704
3. 3.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1521 (b) 1531 (c) 1681 (d) 1599
4. 4.Which is the correct factorisation of x² + 12x + 35? (a) (x + 5)(x + 7) (b) (x + 1)(x + 11) (c) (x − 5)(x − 7) (d) (x + 1)(x + 35)
5. 5.What are the roots of x² − 5x + 6 = 0? (a) 1 and 4 (b) 2 and 4 (c) 1 and 6 (d) 2 and 3
6. 6.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2525 (b) 3035 (c) 2025 (d) 3025
7. 7.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 225 (c) 9 (d) 3
8. 8.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 1)(x + 5) (c) (x + 2)(x + 4) (d) (x − 2)(x − 4)
9. 9.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4624 (b) 5184 (c) 4896 (d) 4634
10. 10.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4356 (b) 5486 (c) 5476 (d) 4916
11. 11.If a + b = 6 and ab = 8, what is a² + b²? (a) 28 (b) 52 (c) 36 (d) 20
12. 12.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 1)(x + 28) (c) (x + 1)(x + 10) (d) (x + 4)(x + 7)
13. 13.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 904 (b) 1024 (c) 1034 (d) 784
14. 14.If a + b = 15 and ab = 44, what is a² + b²? (a) 137 (b) 181 (c) 313 (d) 225
15. 15.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1691 (b) 1681 (c) 1601 (d) 1521
16. 16.If a + b = 13 and ab = 42, what is (a − b)²? (a) 85 (b) 127 (c) 1 (d) 169
17. 17.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 375 (b) 235 (c) 225 (d) 625
18. 18.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 73 (c) 17 (d) 71
19. 19.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x + 1)(x + 9) (c) (x − 3)(x − 7) (d) (x + 3)(x + 7)
20. 20.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)
21. 21.Which is the correct factorisation of x² + 11x + 24? (a) (x + 3)(x + 8) (b) (x + 1)(x + 24) (c) (x − 3)(x − 8) (d) (x + 1)(x + 10)
22. 22.What are the roots of x² − 11x + 28 = 0? (a) 2 and 14 (b) 1 and 28 (c) 4 and 7 (d) 1 and 10
23. 23.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10609 (b) 10009 (c) 10619 (d) 9409
24. 24.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2035 (b) 1225 (c) 1625 (d) 2025
25. 25.If a + b = 12 and ab = 35, what is a² + b²? (a) 144 (b) 109 (c) 214 (d) 74
26. 26.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 271 (c) 181 (d) 541
27. 27.If a + b = 9 and ab = 18, what is (a − b)²? (a) 9 (b) 81 (c) 3 (d) 45
28. 28.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x − 7)(x − 8) (c) (x + 7)(x + 8) (d) (x + 1)(x + 14)
29. 29.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 5929 (b) 5939 (c) 6391 (d) 6889
30. 30.What are the roots of x² − 10x + 21 = 0? (a) 3 and 7 (b) 2 and 8 (c) 1 and 21 (d) 1 and 9
31. 31.If a + b = 20 and ab = 96, what is a² + b²? (a) 208 (b) 400 (c) 592 (d) 304
32. 32.If a + b = 20 and ab = 99, what is (a − b)²? (a) 4 (b) 400 (c) 202 (d) 2
33. 33.If a + b = 3 and ab = 2, what is a² + b²? (a) 9 (b) 13 (c) 5 (d) 7
34. 34.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1225 (b) 1575 (c) 1235 (d) 2025
35. 35.If a + b = 13 and ab = 30, what is a² + b²? (a) 139 (b) 229 (c) 169 (d) 109
36. 36.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 404 (b) 324 (c) 484 (d) 494
37. 37.If a + b = 13 and ab = 40, what is (a − b)²? (a) 89 (b) 169 (c) 3 (d) 9
38. 38.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x − 2)(x − 5) (c) (x + 1)(x + 10) (d) (x + 2)(x + 5)
39. 39.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 4)(x + 9) (c) (x + 1)(x + 12) (d) (x + 1)(x + 36)
40. 40.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 971 (b) 841 (c) 961 (d) 901

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