Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 931 · 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 − 17x + 72 = 0? (a) 17 (b) 16 (c) 72 (d) 18
2. 2.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1681 (b) 1521 (c) 1599 (d) 1531
3. 3.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4624 (b) 5184 (c) 4896 (d) 4634
4. 4.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2714 (b) 2504 (c) 2704 (d) 2304
5. 5.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 7)(x + 8) (c) (x + 1)(x + 56) (d) (x + 1)(x + 14)
6. 6.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x + 5)(x + 9) (c) (x + 1)(x + 13) (d) (x − 5)(x − 9)
7. 7.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 11) (c) (x + 4)(x + 8) (d) (x + 1)(x + 32)
8. 8.If a + b = 20 and ab = 99, what is a² + b²? (a) 202 (b) 301 (c) 400 (d) 598
9. 9.If a + b = 12 and ab = 35, what is a² + b²? (a) 74 (b) 109 (c) 144 (d) 214
10. 10.If a + b = 13 and ab = 36, what is a² + b²? (a) 241 (b) 169 (c) 97 (d) 133
11. 11.What is the product of the roots of x² − 9x + 14 = 0? (a) 14 (b) 15 (c) 13 (d) 9
12. 12.What is the sum of the roots of x² − 11x + 18 = 0? (a) 12 (b) 18 (c) 10 (d) 11
13. 13.What is the sum of the roots of x² − 11x + 28 = 0? (a) 11 (b) 12 (c) 10 (d) 28
14. 14.If a + b = 15 and ab = 54, what is (a − b)²? (a) 117 (b) 225 (c) 9 (d) 3
15. 15.What are the roots of x² − 15x + 56 = 0? (a) 1 and 14 (b) 2 and 28 (c) 1 and 56 (d) 7 and 8
16. 16.What are the roots of x² − 14x + 45 = 0? (a) 1 and 13 (b) 3 and 15 (c) 1 and 45 (d) 5 and 9
17. 17.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7066 (b) 6416 (c) 7056 (d) 5776
18. 18.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 16) (c) (x + 1)(x + 72) (d) (x − 8)(x − 9)
19. 19.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 1)(x + 15) (c) (x − 7)(x − 9) (d) (x + 7)(x + 9)
20. 20.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 729 (b) 1089 (c) 1099 (d) 909
21. 21.If a + b = 11 and ab = 28, what is (a − b)²? (a) 3 (b) 121 (c) 9 (d) 65
22. 22.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 925 (b) 1235 (c) 625 (d) 1225
23. 23.If a + b = 15 and ab = 44, what is (a − b)²? (a) 225 (b) 7 (c) 49 (d) 137
24. 24.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 15 (c) 16 (d) 14
25. 25.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 1)(x + 14) (c) (x − 6)(x − 9) (d) (x + 6)(x + 9)
26. 26.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 784 (b) 1024 (c) 896 (d) 794
27. 27.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x + 4)(x + 6) (c) (x − 4)(x − 6) (d) (x + 1)(x + 9)
28. 28.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1764 (b) 1604 (c) 1444 (d) 1774
29. 29.What are the roots of x² − 9x + 14 = 0? (a) 2 and 7 (b) 3 and 6 (c) 1 and 14 (d) 1 and 8
30. 30.If a + b = 12 and ab = 32, what is (a − b)²? (a) 80 (b) 4 (c) 16 (d) 144
31. 31.If a + b = 10 and ab = 9, what is a² + b²? (a) 118 (b) 91 (c) 82 (d) 100
32. 32.What is the sum of the roots of x² − 13x + 36 = 0? (a) 14 (b) 36 (c) 13 (d) 12
33. 33.If a + b = 7 and ab = 6, what is (a − b)²? (a) 49 (b) 25 (c) 37 (d) 5
34. 34.If a + b = 16 and ab = 55, what is a² + b²? (a) 146 (b) 201 (c) 256 (d) 366
35. 35.What are the roots of x² − 12x + 27 = 0? (a) 3 and 9 (b) 1 and 11 (c) 2 and 10 (d) 1 and 27
36. 36.What is the sum of the roots of x² − 6x + 8 = 0? (a) 6 (b) 5 (c) 7 (d) 8
37. 37.If a + b = 10 and ab = 21, what is (a − b)²? (a) 58 (b) 4 (c) 16 (d) 100
38. 38.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 145 (c) 217 (d) 289
39. 39.If a + b = 9 and ab = 20, what is (a − b)²? (a) 81 (b) 1 (c) 61 (d) 41
40. 40.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 5)(x + 7) (c) (x + 1)(x + 35) (d) (x + 1)(x + 11)

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