Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 422 · 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.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1089 (b) 1099 (c) 729 (d) 909
2. 2.If a + b = 14 and ab = 24, what is a² + b²? (a) 244 (b) 172 (c) 196 (d) 148
3. 3.What is the product of the roots of x² − 11x + 30 = 0? (a) 11 (b) 31 (c) 29 (d) 30
4. 4.What is the sum of the roots of x² − 9x + 14 = 0? (a) 14 (b) 8 (c) 10 (d) 9
5. 5.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x − 8)(x − 9) (c) (x + 8)(x + 9) (d) (x + 1)(x + 16)
6. 6.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2475 (b) 2025 (c) 2035 (d) 3025
7. 7.If a + b = 12 and ab = 20, what is (a − b)²? (a) 144 (b) 64 (c) 104 (d) 8
8. 8.What is the product of the roots of x² − 10x + 24 = 0? (a) 25 (b) 10 (c) 24 (d) 23
9. 9.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6889 (b) 6899 (c) 6409 (d) 5929
10. 10.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x + 1)(x + 8) (c) (x − 2)(x − 4) (d) (x + 2)(x + 4)
11. 11.If a + b = 16 and ab = 63, what is (a − b)²? (a) 130 (b) 256 (c) 4 (d) 2
12. 12.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 17 (c) 63 (d) 15
13. 13.If a + b = 18 and ab = 77, what is (a − b)²? (a) 324 (b) 170 (c) 16 (d) 4
14. 14.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x − 6)(x − 9) (c) (x + 1)(x + 14) (d) (x + 1)(x + 54)
15. 15.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1591 (b) 1379 (c) 1849 (d) 1369
16. 16.Which is the correct factorisation of x² + 12x + 27? (a) (x + 3)(x + 9) (b) (x + 1)(x + 11) (c) (x − 3)(x − 9) (d) (x + 1)(x + 27)
17. 17.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 404 (b) 324 (c) 494 (d) 484
18. 18.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 32) (c) (x + 4)(x + 8) (d) (x + 1)(x + 11)
19. 19.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8109 (b) 8659 (c) 8649 (d) 7569
20. 20.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)
21. 21.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 1)(x + 45) (c) (x + 5)(x + 9) (d) (x + 1)(x + 13)
22. 22.If a + b = 8 and ab = 12, what is a² + b²? (a) 40 (b) 64 (c) 52 (d) 88
23. 23.If a + b = 16 and ab = 60, what is a² + b²? (a) 136 (b) 256 (c) 196 (d) 376
24. 24.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x − 5)(x − 6) (c) (x + 1)(x + 10) (d) (x + 1)(x + 30)
25. 25.If a + b = 13 and ab = 36, what is (a − b)²? (a) 97 (b) 25 (c) 169 (d) 5
26. 26.If a + b = 15 and ab = 56, what is (a − b)²? (a) 225 (b) 113 (c) 1 (d) 169
27. 27.What are the roots of x² − 9x + 14 = 0? (a) 3 and 6 (b) 1 and 8 (c) 2 and 7 (d) 1 and 14
28. 28.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 6)(x + 7) (c) (x − 6)(x − 7) (d) (x + 1)(x + 42)
29. 29.What are the roots of x² − 10x + 16 = 0? (a) 1 and 9 (b) 1 and 16 (c) 2 and 8 (d) 3 and 7
30. 30.Which is the correct factorisation of x² + 10x + 16? (a) (x + 2)(x + 8) (b) (x + 1)(x + 16) (c) (x − 2)(x − 8) (d) (x + 1)(x + 9)
31. 31.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2411 (b) 2499 (c) 2401 (d) 2601
32. 32.If a + b = 9 and ab = 20, what is a² + b²? (a) 81 (b) 61 (c) 41 (d) 121
33. 33.If a + b = 8 and ab = 15, what is a² + b²? (a) 64 (b) 49 (c) 94 (d) 34
34. 34.If a + b = 18 and ab = 72, what is a² + b²? (a) 468 (b) 252 (c) 324 (d) 180
35. 35.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 416 (b) 256 (c) 586 (d) 576
36. 36.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 1)(x + 18) (c) (x + 1)(x + 8) (d) (x + 3)(x + 6)
37. 37.If a + b = 11 and ab = 18, what is (a − b)²? (a) 7 (b) 49 (c) 85 (d) 121
38. 38.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 181 (c) 271 (d) 541
39. 39.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2025 (b) 2525 (c) 3035 (d) 3025
40. 40.If a + b = 13 and ab = 30, what is a² + b²? (a) 229 (b) 139 (c) 169 (d) 109

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