Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 361 · 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.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 8) (b) (x + 2)(x + 7) (c) (x + 1)(x + 14) (d) (x − 2)(x − 7)
2. 2.If a + b = 19 and ab = 88, what is (a − b)²? (a) 185 (b) 361 (c) 9 (d) 3
3. 3.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1379 (b) 1591 (c) 1849 (d) 1369
4. 4.If a + b = 9 and ab = 18, what is (a − b)²? (a) 81 (b) 3 (c) 9 (d) 45
5. 5.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 14) (c) (x + 1)(x + 54) (d) (x + 6)(x + 9)
6. 6.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 576 (b) 416 (c) 586 (d) 256
7. 7.If a + b = 15 and ab = 36, what is a² + b²? (a) 297 (b) 153 (c) 189 (d) 225
8. 8.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 71 (c) 72 (d) 73
9. 9.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 961 (b) 899 (c) 851 (d) 841
10. 10.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 909 (b) 1089 (c) 729 (d) 1099
11. 11.What is the product of the roots of x² − 6x + 8 = 0? (a) 9 (b) 7 (c) 8 (d) 6
12. 12.What is the sum of the roots of x² − 13x + 36 = 0? (a) 14 (b) 12 (c) 13 (d) 36
13. 13.Which is the correct factorisation of x² + 8x + 15? (a) (x + 3)(x + 5) (b) (x − 3)(x − 5) (c) (x + 1)(x + 7) (d) (x + 1)(x + 15)
14. 14.If a + b = 15 and ab = 50, what is (a − b)²? (a) 125 (b) 225 (c) 5 (d) 25
15. 15.If a + b = 14 and ab = 48, what is a² + b²? (a) 148 (b) 292 (c) 100 (d) 196
16. 16.If a + b = 16 and ab = 60, what is a² + b²? (a) 376 (b) 196 (c) 256 (d) 136
17. 17.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4624 (b) 5194 (c) 4904 (d) 5184
18. 18.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x − 7)(x − 9) (c) (x + 1)(x + 63) (d) (x + 1)(x + 15)
19. 19.If a + b = 18 and ab = 77, what is a² + b²? (a) 170 (b) 324 (c) 478 (d) 247
20. 20.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 8 and 9 (c) 1 and 72 (d) 1 and 16
21. 21.If a + b = 9 and ab = 18, what is a² + b²? (a) 63 (b) 81 (c) 45 (d) 117
22. 22.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
23. 23.What are the roots of x² − 15x + 54 = 0? (a) 6 and 9 (b) 2 and 27 (c) 1 and 14 (d) 1 and 54
24. 24.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1859 (b) 1849 (c) 1609 (d) 1369
25. 25.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 1)(x + 48) (c) (x − 6)(x − 8) (d) (x + 6)(x + 8)
26. 26.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x − 4)(x − 7) (c) (x + 4)(x + 7) (d) (x + 1)(x + 10)
27. 27.If a + b = 19 and ab = 90, what is (a − b)²? (a) 361 (b) 1 (c) 271 (d) 181
28. 28.If a + b = 7 and ab = 6, what is (a − b)²? (a) 49 (b) 37 (c) 5 (d) 25
29. 29.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 4)(x + 6) (c) (x + 1)(x + 9) (d) (x + 1)(x + 24)
30. 30.What is the product of the roots of x² − 9x + 18 = 0? (a) 9 (b) 17 (c) 19 (d) 18
31. 31.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x − 2)(x − 3) (c) (x + 2)(x + 3) (d) (x + 1)(x + 6)
32. 32.What is the product of the roots of x² − 9x + 20 = 0? (a) 9 (b) 20 (c) 21 (d) 19
33. 33.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 235 (b) 375 (c) 625 (d) 225
34. 34.If a + b = 11 and ab = 18, what is a² + b²? (a) 157 (b) 121 (c) 85 (d) 103
35. 35.What is the product of the roots of x² − 11x + 24 = 0? (a) 25 (b) 23 (c) 24 (d) 11
36. 36.If a + b = 11 and ab = 10, what is a² + b²? (a) 121 (b) 111 (c) 101 (d) 141
37. 37.If a + b = 17 and ab = 70, what is a² + b²? (a) 289 (b) 429 (c) 149 (d) 219
38. 38.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x + 3)(x + 6) (c) (x − 3)(x − 6) (d) (x + 1)(x + 18)
39. 39.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8649 (b) 7579 (c) 7569 (d) 8091
40. 40.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8474 (b) 8464 (c) 7744 (d) 8104

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