Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 371 · 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 = 13 and ab = 42, what is (a − b)²? (a) 169 (b) 127 (c) 85 (d) 1
2. 2.What are the roots of x² − 9x + 14 = 0? (a) 3 and 6 (b) 1 and 14 (c) 2 and 7 (d) 1 and 8
3. 3.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 225 (c) 441 (d) 657
4. 4.What are the roots of x² − 9x + 20 = 0? (a) 4 and 5 (b) 2 and 10 (c) 1 and 8 (d) 1 and 20
5. 5.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x − 3)(x − 7) (c) (x + 3)(x + 7) (d) (x + 1)(x + 9)
6. 6.If a + b = 16 and ab = 48, what is a² + b²? (a) 352 (b) 208 (c) 160 (d) 256
7. 7.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4624 (b) 4634 (c) 4896 (d) 5184
8. 8.Which is the correct factorisation of x² + 11x + 18? (a) (x − 2)(x − 9) (b) (x + 1)(x + 18) (c) (x + 1)(x + 10) (d) (x + 2)(x + 9)
9. 9.What is the sum of the roots of x² − 7x + 10 = 0? (a) 7 (b) 6 (c) 8 (d) 10
10. 10.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 5041 (b) 4771 (c) 4899 (d) 4761
11. 11.If a + b = 10 and ab = 16, what is a² + b²? (a) 68 (b) 84 (c) 100 (d) 132
12. 12.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 1089 (b) 729 (c) 891 (d) 739
13. 13.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 1)(x + 15) (c) (x + 7)(x + 9) (d) (x − 7)(x − 9)
14. 14.If a + b = 20 and ab = 99, what is a² + b²? (a) 301 (b) 400 (c) 598 (d) 202
15. 15.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4884 (b) 5476 (c) 4366 (d) 4356
16. 16.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1225 (b) 2025 (c) 1625 (d) 2035
17. 17.If a + b = 5 and ab = 6, what is (a − b)²? (a) 19 (b) 13 (c) 1 (d) 25
18. 18.If a + b = 13 and ab = 40, what is (a − b)²? (a) 3 (b) 169 (c) 9 (d) 89
19. 19.If a + b = 13 and ab = 12, what is (a − b)²? (a) 145 (b) 11 (c) 169 (d) 121
20. 20.What are the roots of x² − 11x + 30 = 0? (a) 1 and 30 (b) 2 and 15 (c) 5 and 6 (d) 1 and 10
21. 21.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x + 1)(x + 45) (c) (x − 5)(x − 9) (d) (x + 5)(x + 9)
22. 22.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3364 (b) 3604 (c) 3844 (d) 3854
23. 23.If a + b = 12 and ab = 35, what is a² + b²? (a) 74 (b) 109 (c) 214 (d) 144
24. 24.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3364 (b) 3374 (c) 3596 (d) 3844
25. 25.If a + b = 11 and ab = 30, what is (a − b)²? (a) 61 (b) 121 (c) 91 (d) 1
26. 26.What is the product of the roots of x² − 15x + 54 = 0? (a) 15 (b) 55 (c) 53 (d) 54
27. 27.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 73 (c) 17 (d) 72
28. 28.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 841 (b) 961 (c) 971 (d) 901
29. 29.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 416 (c) 586 (d) 576
30. 30.If a + b = 19 and ab = 84, what is (a − b)²? (a) 5 (b) 25 (c) 193 (d) 361
31. 31.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3249 (b) 3969 (c) 3979 (d) 3609
32. 32.What are the roots of x² − 7x + 12 = 0? (a) 1 and 6 (b) 3 and 4 (c) 1 and 12 (d) 2 and 6
33. 33.What is the sum of the roots of x² − 13x + 42 = 0? (a) 14 (b) 12 (c) 13 (d) 42
34. 34.What is the sum of the roots of x² − 12x + 27 = 0? (a) 11 (b) 12 (c) 13 (d) 27
35. 35.Which is the correct factorisation of x² + 11x + 24? (a) (x − 3)(x − 8) (b) (x + 3)(x + 8) (c) (x + 1)(x + 24) (d) (x + 1)(x + 10)
36. 36.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 5184 (b) 4904 (c) 5194 (d) 4624
37. 37.If a + b = 19 and ab = 90, what is a² + b²? (a) 181 (b) 361 (c) 271 (d) 541
38. 38.If a + b = 12 and ab = 11, what is a² + b²? (a) 133 (b) 166 (c) 144 (d) 122
39. 39.What is the sum of the roots of x² − 10x + 24 = 0? (a) 24 (b) 10 (c) 9 (d) 11
40. 40.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1235 (b) 1575 (c) 1225 (d) 2025

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