Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 879 · 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 = 8 and ab = 15, what is a² + b²? (a) 94 (b) 49 (c) 34 (d) 64
2. 2.If a + b = 10 and ab = 16, what is (a − b)²? (a) 68 (b) 6 (c) 36 (d) 100
3. 3.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3625 (b) 4235 (c) 4225 (d) 3025
4. 4.If a + b = 6 and ab = 5, what is a² + b²? (a) 31 (b) 26 (c) 36 (d) 46
5. 5.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2126 (b) 2484 (c) 2116 (d) 2916
6. 6.If a + b = 15 and ab = 44, what is a² + b²? (a) 137 (b) 225 (c) 181 (d) 313
7. 7.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 301 (c) 202 (d) 400
8. 8.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 2 and 36 (c) 1 and 72 (d) 1 and 16
9. 9.If a + b = 15 and ab = 36, what is (a − b)²? (a) 153 (b) 9 (c) 225 (d) 81
10. 10.If a + b = 9 and ab = 18, what is a² + b²? (a) 63 (b) 117 (c) 45 (d) 81
11. 11.What is the product of the roots of x² − 5x + 6 = 0? (a) 7 (b) 8 (c) 5 (d) 6
12. 12.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2501 (b) 2611 (c) 2601 (d) 2401
13. 13.What are the roots of x² − 11x + 28 = 0? (a) 1 and 10 (b) 1 and 28 (c) 2 and 14 (d) 4 and 7
14. 14.If a + b = 18 and ab = 77, what is a² + b²? (a) 170 (b) 247 (c) 324 (d) 478
15. 15.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3035 (b) 3025 (c) 3575 (d) 4225
16. 16.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1625 (b) 2035 (c) 2025 (d) 1225
17. 17.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 217 (c) 289 (d) 145
18. 18.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x − 2)(x − 7) (c) (x + 2)(x + 7) (d) (x + 1)(x + 8)
19. 19.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 5184 (b) 4624 (c) 4634 (d) 4896
20. 20.What is the product of the roots of x² − 9x + 14 = 0? (a) 14 (b) 13 (c) 9 (d) 15
21. 21.What is the product of the roots of x² − 13x + 42 = 0? (a) 42 (b) 41 (c) 43 (d) 13
22. 22.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x − 6)(x − 7) (c) (x + 6)(x + 7) (d) (x + 1)(x + 42)
23. 23.If a + b = 13 and ab = 12, what is (a − b)²? (a) 11 (b) 145 (c) 169 (d) 121
24. 24.What is the sum of the roots of x² − 14x + 45 = 0? (a) 14 (b) 45 (c) 15 (d) 13
25. 25.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 7225 (b) 9035 (c) 9025 (d) 8125
26. 26.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 17 (c) 72 (d) 16
27. 27.What are the roots of x² − 16x + 63 = 0? (a) 3 and 21 (b) 1 and 63 (c) 1 and 15 (d) 7 and 9
28. 28.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6094 (b) 6724 (c) 6396 (d) 6084
29. 29.If a + b = 15 and ab = 54, what is (a − b)²? (a) 3 (b) 225 (c) 117 (d) 9
30. 30.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8464 (b) 7754 (c) 8096 (d) 7744
31. 31.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 72) (c) (x + 1)(x + 16) (d) (x − 8)(x − 9)
32. 32.If a + b = 6 and ab = 8, what is a² + b²? (a) 28 (b) 36 (c) 20 (d) 52
33. 33.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6404 (b) 6724 (c) 6084 (d) 6734
34. 34.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8291 (b) 8101 (c) 7921 (d) 8281
35. 35.What are the roots of x² − 13x + 36 = 0? (a) 1 and 12 (b) 4 and 9 (c) 1 and 36 (d) 2 and 18
36. 36.If a + b = 7 and ab = 10, what is a² + b²? (a) 29 (b) 39 (c) 49 (d) 69
37. 37.What are the roots of x² − 11x + 30 = 0? (a) 1 and 10 (b) 1 and 30 (c) 2 and 15 (d) 5 and 6
38. 38.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x − 5)(x − 7) (c) (x + 1)(x + 35) (d) (x + 5)(x + 7)
39. 39.If a + b = 14 and ab = 40, what is a² + b²? (a) 116 (b) 276 (c) 156 (d) 196
40. 40.If a + b = 5 and ab = 4, what is (a − b)²? (a) 25 (b) 3 (c) 17 (d) 9

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