Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 843 · 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 sum of the roots of x² − 13x + 36 = 0? (a) 12 (b) 14 (c) 13 (d) 36
2. 2.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8281 (b) 7921 (c) 8099 (d) 7931
3. 3.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1764 (b) 1774 (c) 1444 (d) 1604
4. 4.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 494 (b) 404 (c) 324 (d) 484
5. 5.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)
6. 6.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 1 and 14 (c) 7 and 8 (d) 2 and 28
7. 7.What is the product of the roots of x² − 11x + 30 = 0? (a) 30 (b) 11 (c) 29 (d) 31
8. 8.If a + b = 14 and ab = 45, what is a² + b²? (a) 286 (b) 196 (c) 106 (d) 151
9. 9.If a + b = 6 and ab = 5, what is (a − b)²? (a) 16 (b) 4 (c) 36 (d) 26
10. 10.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 409 (b) 529 (c) 539 (d) 289
11. 11.If a + b = 15 and ab = 50, what is (a − b)²? (a) 125 (b) 225 (c) 25 (d) 5
12. 12.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 1)(x + 30) (c) (x − 5)(x − 6) (d) (x + 5)(x + 6)
13. 13.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 16 (d) 17
14. 14.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 8 and 9 (c) 2 and 36 (d) 1 and 16
15. 15.If a + b = 11 and ab = 24, what is a² + b²? (a) 169 (b) 73 (c) 121 (d) 97
16. 16.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4624 (b) 4904 (c) 5184 (d) 5194
17. 17.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3491 (b) 3599 (c) 3721 (d) 3481
18. 18.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 9 (c) 3 (d) 225
19. 19.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 6)(x + 8) (c) (x − 6)(x − 8) (d) (x + 1)(x + 48)
20. 20.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6734 (b) 6724 (c) 6084 (d) 6404
21. 21.What is the sum of the roots of x² − 14x + 45 = 0? (a) 15 (b) 13 (c) 14 (d) 45
22. 22.If a + b = 14 and ab = 40, what is a² + b²? (a) 196 (b) 116 (c) 276 (d) 156
23. 23.If a + b = 6 and ab = 8, what is a² + b²? (a) 36 (b) 52 (c) 28 (d) 20
24. 24.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 11) (b) (x + 1)(x + 32) (c) (x − 4)(x − 8) (d) (x + 4)(x + 8)
25. 25.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 15) (c) (x + 7)(x + 9) (d) (x + 1)(x + 63)
26. 26.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 5625 (b) 5635 (c) 4925 (d) 4225
27. 27.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 42) (c) (x + 1)(x + 12) (d) (x + 6)(x + 7)
28. 28.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 901 (b) 841 (c) 961 (d) 971
29. 29.What is the sum of the roots of x² − 9x + 18 = 0? (a) 10 (b) 9 (c) 18 (d) 8
30. 30.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 361 (b) 451 (c) 441 (d) 401
31. 31.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8281 (b) 7921 (c) 8291 (d) 8101
32. 32.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 7225 (b) 5635 (c) 6375 (d) 5625
33. 33.Which is the correct factorisation of x² + 11x + 28? (a) (x + 4)(x + 7) (b) (x + 1)(x + 28) (c) (x − 4)(x − 7) (d) (x + 1)(x + 10)
34. 34.Which is the correct factorisation of x² + 7x + 10? (a) (x − 2)(x − 5) (b) (x + 2)(x + 5) (c) (x + 1)(x + 10) (d) (x + 1)(x + 6)
35. 35.If a + b = 9 and ab = 8, what is (a − b)²? (a) 49 (b) 81 (c) 65 (d) 7
36. 36.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4771 (b) 4761 (c) 4899 (d) 5041
37. 37.What is the product of the roots of x² − 9x + 14 = 0? (a) 13 (b) 9 (c) 14 (d) 15
38. 38.If a + b = 9 and ab = 20, what is a² + b²? (a) 81 (b) 41 (c) 61 (d) 121
39. 39.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x − 6)(x − 9) (c) (x + 1)(x + 54) (d) (x + 1)(x + 14)
40. 40.If a + b = 12 and ab = 20, what is a² + b²? (a) 124 (b) 184 (c) 104 (d) 144

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