Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 625 · 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 − 11x + 30 = 0? (a) 12 (b) 30 (c) 10 (d) 11
2. 2.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3364 (b) 3604 (c) 3854 (d) 3844
3. 3.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)
4. 4.If a + b = 13 and ab = 22, what is a² + b²? (a) 169 (b) 125 (c) 147 (d) 213
5. 5.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 1)(x + 63) (c) (x + 7)(x + 9) (d) (x − 7)(x − 9)
6. 6.If a + b = 9 and ab = 18, what is (a − b)²? (a) 9 (b) 45 (c) 3 (d) 81
7. 7.What is the product of the roots of x² − 12x + 35 = 0? (a) 12 (b) 34 (c) 36 (d) 35
8. 8.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 17 (c) 15 (d) 16
9. 9.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x + 4)(x + 9) (c) (x − 4)(x − 9) (d) (x + 1)(x + 12)
10. 10.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x − 7)(x − 8) (c) (x + 7)(x + 8) (d) (x + 1)(x + 56)
11. 11.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3601 (b) 3721 (c) 3731 (d) 3481
12. 12.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 14 (c) 13 (d) 15
13. 13.If a + b = 14 and ab = 40, what is (a − b)²? (a) 6 (b) 36 (c) 116 (d) 196
14. 14.If a + b = 8 and ab = 12, what is (a − b)²? (a) 4 (b) 16 (c) 64 (d) 40
15. 15.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x + 1)(x + 12) (c) (x + 2)(x + 6) (d) (x − 2)(x − 6)
16. 16.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 909 (b) 1089 (c) 729 (d) 1099
17. 17.What is the sum of the roots of x² − 5x + 6 = 0? (a) 6 (b) 4 (c) 5 (d) 7
18. 18.Which is the correct factorisation of x² + 11x + 18? (a) (x + 2)(x + 9) (b) (x + 1)(x + 10) (c) (x + 1)(x + 18) (d) (x − 2)(x − 9)
19. 19.If a + b = 12 and ab = 35, what is a² + b²? (a) 214 (b) 144 (c) 74 (d) 109
20. 20.If a + b = 14 and ab = 40, what is a² + b²? (a) 196 (b) 116 (c) 276 (d) 156
21. 21.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x + 1)(x + 16)
22. 22.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6724 (b) 6094 (c) 6396 (d) 6084
23. 23.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1225 (b) 1235 (c) 625 (d) 925
24. 24.If a + b = 9 and ab = 20, what is a² + b²? (a) 121 (b) 81 (c) 41 (d) 61
25. 25.What is the product of the roots of x² − 13x + 40 = 0? (a) 13 (b) 41 (c) 39 (d) 40
26. 26.If a + b = 18 and ab = 80, what is (a − b)²? (a) 4 (b) 324 (c) 2 (d) 164
27. 27.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 4)(x + 6) (c) (x + 1)(x + 24) (d) (x + 1)(x + 9)
28. 28.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1936 (b) 1584 (c) 1296 (d) 1306
29. 29.What is the sum of the roots of x² − 14x + 45 = 0? (a) 15 (b) 45 (c) 13 (d) 14
30. 30.If a + b = 16 and ab = 48, what is (a − b)²? (a) 160 (b) 256 (c) 8 (d) 64
31. 31.If a + b = 16 and ab = 60, what is a² + b²? (a) 376 (b) 256 (c) 136 (d) 196
32. 32.What is the product of the roots of x² − 10x + 24 = 0? (a) 10 (b) 23 (c) 24 (d) 25
33. 33.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 4106 (b) 3616 (c) 4096 (d) 3136
34. 34.If a + b = 10 and ab = 21, what is (a − b)²? (a) 16 (b) 58 (c) 4 (d) 100
35. 35.If a + b = 15 and ab = 36, what is (a − b)²? (a) 153 (b) 81 (c) 225 (d) 9
36. 36.If a + b = 19 and ab = 88, what is (a − b)²? (a) 361 (b) 185 (c) 3 (d) 9
37. 37.If a + b = 12 and ab = 27, what is a² + b²? (a) 198 (b) 117 (c) 144 (d) 90
38. 38.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1369 (b) 1591 (c) 1849 (d) 1379
39. 39.If a + b = 5 and ab = 6, what is a² + b²? (a) 37 (b) 19 (c) 25 (d) 13
40. 40.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 9025 (b) 7235 (c) 8075 (d) 7225

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