Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 754 · 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 79²? (a) 6561 (b) 6251 (c) 6399 (d) 6241
2. 2.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 5476 (b) 5486 (c) 4916 (d) 4356
3. 3.What is the product of the roots of x² − 9x + 18 = 0? (a) 9 (b) 18 (c) 17 (d) 19
4. 4.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 484 (b) 404 (c) 324 (d) 494
5. 5.What is the sum of the roots of x² − 9x + 18 = 0? (a) 10 (b) 18 (c) 9 (d) 8
6. 6.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3364 (b) 3604 (c) 3854 (d) 3844
7. 7.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 925 (b) 625 (c) 1225 (d) 1235
8. 8.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 299 (b) 529 (c) 289 (d) 391
9. 9.If a + b = 5 and ab = 6, what is (a − b)²? (a) 13 (b) 25 (c) 19 (d) 1
10. 10.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6084 (b) 6094 (c) 6396 (d) 6724
11. 11.If a + b = 10 and ab = 16, what is (a − b)²? (a) 6 (b) 68 (c) 100 (d) 36
12. 12.What are the roots of x² − 8x + 15 = 0? (a) 3 and 5 (b) 2 and 6 (c) 1 and 7 (d) 1 and 15
13. 13.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 72 (c) 16 (d) 17
14. 14.If a + b = 6 and ab = 5, what is (a − b)²? (a) 16 (b) 4 (c) 26 (d) 36
15. 15.What is the sum of the roots of x² − 10x + 21 = 0? (a) 10 (b) 9 (c) 11 (d) 21
16. 16.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 7056 (b) 5786 (c) 6384 (d) 5776
17. 17.What is the sum of the roots of x² − 12x + 35 = 0? (a) 13 (b) 12 (c) 11 (d) 35
18. 18.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x − 4)(x − 9) (c) (x + 4)(x + 9) (d) (x + 1)(x + 36)
19. 19.What is the sum of the roots of x² − 9x + 20 = 0? (a) 9 (b) 20 (c) 8 (d) 10
20. 20.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 56) (c) (x − 7)(x − 8) (d) (x + 1)(x + 14)
21. 21.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1521 (b) 1601 (c) 1691 (d) 1681
22. 22.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4235 (b) 4875 (c) 4225 (d) 5625
23. 23.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 40) (c) (x + 1)(x + 12) (d) (x + 5)(x + 8)
24. 24.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6571 (b) 6561 (c) 6241 (d) 6401
25. 25.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 7)(x + 9) (c) (x − 7)(x − 9) (d) (x + 1)(x + 15)
26. 26.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2501 (b) 2611 (c) 2401 (d) 2601
27. 27.If a + b = 14 and ab = 45, what is (a − b)²? (a) 4 (b) 106 (c) 196 (d) 16
28. 28.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x − 4)(x − 5) (c) (x + 1)(x + 8) (d) (x + 4)(x + 5)
29. 29.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x + 1)(x + 13) (c) (x + 6)(x + 8) (d) (x − 6)(x − 8)
30. 30.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 10609 (b) 9991 (c) 9419 (d) 9409
31. 31.If a + b = 16 and ab = 63, what is a² + b²? (a) 193 (b) 382 (c) 256 (d) 130
32. 32.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 4235 (b) 3625 (c) 3025 (d) 4225
33. 33.If a + b = 13 and ab = 40, what is (a − b)²? (a) 9 (b) 3 (c) 169 (d) 89
34. 34.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6734 (b) 6724 (c) 6404 (d) 6084
35. 35.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 16) (c) (x − 8)(x − 9) (d) (x + 1)(x + 72)
36. 36.If a + b = 13 and ab = 36, what is a² + b²? (a) 133 (b) 169 (c) 241 (d) 97
37. 37.If a + b = 15 and ab = 56, what is a² + b²? (a) 169 (b) 113 (c) 337 (d) 225
38. 38.What are the roots of x² − 13x + 36 = 0? (a) 1 and 36 (b) 2 and 18 (c) 1 and 12 (d) 4 and 9
39. 39.If a + b = 17 and ab = 70, what is (a − b)²? (a) 9 (b) 289 (c) 3 (d) 149
40. 40.If a + b = 20 and ab = 96, what is a² + b²? (a) 400 (b) 592 (c) 304 (d) 208

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