Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 193 · 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 = 14 and ab = 24, what is a² + b²? (a) 196 (b) 244 (c) 172 (d) 148
2. 2.What is the product of the roots of x² − 11x + 28 = 0? (a) 27 (b) 11 (c) 28 (d) 29
3. 3.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 55 (c) 15 (d) 56
4. 4.What are the roots of x² − 13x + 42 = 0? (a) 6 and 7 (b) 2 and 21 (c) 1 and 42 (d) 1 and 12
5. 5.What is the sum of the roots of x² − 9x + 14 = 0? (a) 10 (b) 14 (c) 8 (d) 9
6. 6.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3604 (b) 3844 (c) 3854 (d) 3364
7. 7.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1089 (b) 909 (c) 1099 (d) 729
8. 8.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2314 (b) 2496 (c) 2304 (d) 2704
9. 9.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)
10. 10.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x + 7)(x + 8) (d) (x − 7)(x − 8)
11. 11.If a + b = 12 and ab = 35, what is (a − b)²? (a) 144 (b) 4 (c) 74 (d) 2
12. 12.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 1)(x + 5) (c) (x + 1)(x + 8) (d) (x + 2)(x + 4)
13. 13.If a + b = 14 and ab = 33, what is a² + b²? (a) 262 (b) 130 (c) 196 (d) 163
14. 14.If a + b = 13 and ab = 42, what is a² + b²? (a) 253 (b) 85 (c) 127 (d) 169
15. 15.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x − 4)(x − 5) (c) (x + 4)(x + 5) (d) (x + 1)(x + 8)
16. 16.What are the roots of x² − 15x + 54 = 0? (a) 1 and 14 (b) 1 and 54 (c) 6 and 9 (d) 2 and 27
17. 17.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 334 (b) 396 (c) 324 (d) 484
18. 18.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1625 (b) 1225 (c) 2025 (d) 2035
19. 19.If a + b = 8 and ab = 7, what is a² + b²? (a) 64 (b) 78 (c) 50 (d) 57
20. 20.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 27) (b) (x + 3)(x + 9) (c) (x − 3)(x − 9) (d) (x + 1)(x + 11)
21. 21.What is the sum of the roots of x² − 8x + 15 = 0? (a) 15 (b) 7 (c) 9 (d) 8
22. 22.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 16 (c) 17 (d) 72
23. 23.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 5041 (b) 4771 (c) 4761 (d) 4899
24. 24.If a + b = 17 and ab = 66, what is a² + b²? (a) 157 (b) 421 (c) 289 (d) 223
25. 25.Which is the correct factorisation of x² + 10x + 16? (a) (x + 2)(x + 8) (b) (x + 1)(x + 9) (c) (x − 2)(x − 8) (d) (x + 1)(x + 16)
26. 26.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 441 (b) 401 (c) 451 (d) 361
27. 27.What are the roots of x² − 14x + 48 = 0? (a) 2 and 24 (b) 6 and 8 (c) 1 and 48 (d) 1 and 13
28. 28.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3616 (b) 4106 (c) 4096 (d) 3136
29. 29.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 1 and 72 (c) 2 and 36 (d) 8 and 9
30. 30.If a + b = 9 and ab = 20, what is a² + b²? (a) 121 (b) 61 (c) 81 (d) 41
31. 31.What are the roots of x² − 15x + 56 = 0? (a) 7 and 8 (b) 2 and 28 (c) 1 and 14 (d) 1 and 56
32. 32.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1681 (b) 1599 (c) 1531 (d) 1521
33. 33.If a + b = 9 and ab = 14, what is (a − b)²? (a) 25 (b) 81 (c) 5 (d) 53
34. 34.What is the product of the roots of x² − 13x + 42 = 0? (a) 13 (b) 41 (c) 43 (d) 42
35. 35.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 181 (c) 361 (d) 271
36. 36.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x + 1)(x + 10) (c) (x − 3)(x − 8) (d) (x + 3)(x + 8)
37. 37.What is the product of the roots of x² − 15x + 54 = 0? (a) 15 (b) 55 (c) 54 (d) 53
38. 38.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4489 (b) 4499 (c) 4891 (d) 5329
39. 39.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1454 (b) 1764 (c) 1444 (d) 1596
40. 40.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1774 (b) 1444 (c) 1604 (d) 1764

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