Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 518 · 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 = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 361 (c) 1 (d) 181
2. 2.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1764 (b) 1454 (c) 1596 (d) 1444
3. 3.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 56) (c) (x − 7)(x − 8) (d) (x + 1)(x + 14)
4. 4.Which is the correct factorisation of x² + 10x + 21? (a) (x + 3)(x + 7) (b) (x − 3)(x − 7) (c) (x + 1)(x + 21) (d) (x + 1)(x + 9)
5. 5.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3844 (b) 3596 (c) 3364 (d) 3374
6. 6.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x + 1)(x + 18) (c) (x − 2)(x − 9) (d) (x + 2)(x + 9)
7. 7.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 384 (b) 576 (c) 266 (d) 256
8. 8.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8846 (b) 7396 (c) 8836 (d) 8116
9. 9.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 324 (b) 494 (c) 404 (d) 484
10. 10.If a + b = 14 and ab = 48, what is (a − b)²? (a) 100 (b) 2 (c) 196 (d) 4
11. 11.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x + 2)(x + 3) (c) (x + 1)(x + 4) (d) (x − 2)(x − 3)
12. 12.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3604 (b) 3364 (c) 3854 (d) 3844
13. 13.If a + b = 16 and ab = 63, what is (a − b)²? (a) 256 (b) 2 (c) 130 (d) 4
14. 14.What is the sum of the roots of x² − 13x + 36 = 0? (a) 14 (b) 13 (c) 12 (d) 36
15. 15.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 11) (b) (x + 4)(x + 8) (c) (x + 1)(x + 32) (d) (x − 4)(x − 8)
16. 16.What are the roots of x² − 11x + 18 = 0? (a) 2 and 9 (b) 3 and 6 (c) 1 and 18 (d) 1 and 10
17. 17.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 401 (b) 361 (c) 441 (d) 451
18. 18.If a + b = 13 and ab = 42, what is a² + b²? (a) 85 (b) 169 (c) 253 (d) 127
19. 19.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3136 (b) 4096 (c) 3616 (d) 4106
20. 20.What is the sum of the roots of x² − 7x + 12 = 0? (a) 8 (b) 7 (c) 12 (d) 6
21. 21.If a + b = 17 and ab = 70, what is (a − b)²? (a) 9 (b) 3 (c) 289 (d) 149
22. 22.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 4235 (b) 3625 (c) 3025 (d) 4225
23. 23.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 729 (b) 1089 (c) 909 (d) 1099
24. 24.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10609 (b) 10009 (c) 10619 (d) 9409
25. 25.What is the sum of the roots of x² − 11x + 24 = 0? (a) 12 (b) 11 (c) 10 (d) 24
26. 26.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 12) (c) (x + 1)(x + 6) (d) (x + 3)(x + 4)
27. 27.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)
28. 28.If a + b = 14 and ab = 45, what is a² + b²? (a) 196 (b) 151 (c) 106 (d) 286
29. 29.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x + 6)(x + 9) (c) (x − 6)(x − 9) (d) (x + 1)(x + 54)
30. 30.If a + b = 7 and ab = 6, what is a² + b²? (a) 43 (b) 49 (c) 37 (d) 61
31. 31.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)
32. 32.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6401 (b) 6241 (c) 6561 (d) 6571
33. 33.If a + b = 14 and ab = 24, what is (a − b)²? (a) 196 (b) 148 (c) 10 (d) 100
34. 34.If a + b = 12 and ab = 20, what is a² + b²? (a) 144 (b) 124 (c) 184 (d) 104
35. 35.Which is the correct factorisation of x² + 9x + 18? (a) (x + 3)(x + 6) (b) (x + 1)(x + 8) (c) (x − 3)(x − 6) (d) (x + 1)(x + 18)
36. 36.Which is the correct factorisation of x² + 11x + 24? (a) (x + 3)(x + 8) (b) (x + 1)(x + 10) (c) (x − 3)(x − 8) (d) (x + 1)(x + 24)
37. 37.What is the product of the roots of x² − 15x + 54 = 0? (a) 55 (b) 53 (c) 15 (d) 54
38. 38.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 4)(x + 5) (c) (x + 1)(x + 20) (d) (x + 1)(x + 8)
39. 39.If a + b = 9 and ab = 14, what is (a − b)²? (a) 5 (b) 81 (c) 25 (d) 53
40. 40.If a + b = 19 and ab = 84, what is (a − b)²? (a) 5 (b) 193 (c) 361 (d) 25

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