Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 452 · 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.What is the sum of the roots of x² − 10x + 24 = 0? (a) 11 (b) 24 (c) 10 (d) 9
2. 2.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 15 (c) 16 (d) 56
3. 3.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x + 1)(x + 12) (c) (x − 4)(x − 9) (d) (x + 4)(x + 9)
4. 4.If a + b = 11 and ab = 18, what is (a − b)²? (a) 7 (b) 85 (c) 49 (d) 121
5. 5.If a + b = 20 and ab = 96, what is a² + b²? (a) 400 (b) 304 (c) 592 (d) 208
6. 6.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2916 (b) 2484 (c) 2126 (d) 2116
7. 7.What is the sum of the roots of x² − 14x + 45 = 0? (a) 15 (b) 13 (c) 14 (d) 45
8. 8.If a + b = 19 and ab = 84, what is (a − b)²? (a) 25 (b) 193 (c) 5 (d) 361
9. 9.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 884 (b) 676 (c) 686 (d) 1156
10. 10.What is the sum of the roots of x² − 13x + 42 = 0? (a) 13 (b) 12 (c) 14 (d) 42
11. 11.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1296 (b) 1946 (c) 1616 (d) 1936
12. 12.What is the product of the roots of x² − 12x + 27 = 0? (a) 28 (b) 27 (c) 12 (d) 26
13. 13.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 18) (b) (x − 2)(x − 9) (c) (x + 2)(x + 9) (d) (x + 1)(x + 10)
14. 14.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 529 (b) 289 (c) 391 (d) 299
15. 15.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 8)(x + 9) (c) (x + 1)(x + 16) (d) (x − 8)(x − 9)
16. 16.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 1)(x + 8) (c) (x + 4)(x + 5) (d) (x + 1)(x + 20)
17. 17.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x + 1)(x + 13) (c) (x − 5)(x − 9) (d) (x + 5)(x + 9)
18. 18.What is the product of the roots of x² − 13x + 42 = 0? (a) 41 (b) 42 (c) 13 (d) 43
19. 19.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 4)(x + 6) (c) (x + 1)(x + 24) (d) (x + 1)(x + 9)
20. 20.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 891 (b) 1089 (c) 729 (d) 739
21. 21.What are the roots of x² − 11x + 24 = 0? (a) 3 and 8 (b) 2 and 12 (c) 1 and 10 (d) 1 and 24
22. 22.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4499 (b) 5329 (c) 4891 (d) 4489
23. 23.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 324 (b) 494 (c) 484 (d) 404
24. 24.If a + b = 12 and ab = 32, what is a² + b²? (a) 144 (b) 80 (c) 112 (d) 208
25. 25.What is the sum of the roots of x² − 11x + 28 = 0? (a) 10 (b) 11 (c) 12 (d) 28
26. 26.If a + b = 21 and ab = 108, what is (a − b)²? (a) 9 (b) 441 (c) 3 (d) 225
27. 27.If a + b = 18 and ab = 72, what is a² + b²? (a) 180 (b) 468 (c) 252 (d) 324
28. 28.What are the roots of x² − 5x + 6 = 0? (a) 2 and 3 (b) 2 and 4 (c) 1 and 4 (d) 1 and 6
29. 29.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x + 1)(x + 27) (c) (x − 3)(x − 9) (d) (x + 3)(x + 9)
30. 30.What is the product of the roots of x² − 11x + 30 = 0? (a) 29 (b) 11 (c) 31 (d) 30
31. 31.What are the roots of x² − 13x + 36 = 0? (a) 2 and 18 (b) 1 and 12 (c) 1 and 36 (d) 4 and 9
32. 32.If a + b = 14 and ab = 24, what is (a − b)²? (a) 100 (b) 148 (c) 196 (d) 10
33. 33.If a + b = 15 and ab = 50, what is a² + b²? (a) 125 (b) 225 (c) 175 (d) 325
34. 34.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x + 3)(x + 7) (c) (x + 1)(x + 21) (d) (x − 3)(x − 7)
35. 35.What are the roots of x² − 10x + 16 = 0? (a) 1 and 16 (b) 1 and 9 (c) 3 and 7 (d) 2 and 8
36. 36.What is the sum of the roots of x² − 13x + 40 = 0? (a) 12 (b) 13 (c) 40 (d) 14
37. 37.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 9025 (b) 10025 (c) 11035 (d) 11025
38. 38.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7744 (b) 8464 (c) 8096 (d) 7754
39. 39.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7235 (b) 7225 (c) 8075 (d) 9025
40. 40.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 4)(x + 7) (c) (x + 1)(x + 10) (d) (x + 1)(x + 28)

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