Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 984 · 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.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
2. 2.If a + b = 17 and ab = 70, what is (a − b)²? (a) 9 (b) 149 (c) 289 (d) 3
3. 3.If a + b = 13 and ab = 40, what is (a − b)²? (a) 169 (b) 9 (c) 3 (d) 89
4. 4.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)
5. 5.What is the product of the roots of x² − 10x + 16 = 0? (a) 15 (b) 16 (c) 10 (d) 17
6. 6.If a + b = 14 and ab = 33, what is a² + b²? (a) 196 (b) 262 (c) 130 (d) 163
7. 7.If a + b = 14 and ab = 48, what is (a − b)²? (a) 100 (b) 2 (c) 4 (d) 196
8. 8.What is the sum of the roots of x² − 12x + 27 = 0? (a) 13 (b) 11 (c) 27 (d) 12
9. 9.Which is the correct factorisation of x² + 11x + 18? (a) (x − 2)(x − 9) (b) (x + 1)(x + 10) (c) (x + 2)(x + 9) (d) (x + 1)(x + 18)
10. 10.What is the sum of the roots of x² − 7x + 10 = 0? (a) 10 (b) 6 (c) 7 (d) 8
11. 11.If a + b = 12 and ab = 20, what is (a − b)²? (a) 8 (b) 64 (c) 104 (d) 144
12. 12.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 5476 (b) 4366 (c) 4356 (d) 4884
13. 13.What are the roots of x² − 11x + 28 = 0? (a) 4 and 7 (b) 1 and 10 (c) 2 and 14 (d) 1 and 28
14. 14.What is the product of the roots of x² − 13x + 42 = 0? (a) 43 (b) 13 (c) 42 (d) 41
15. 15.What is the product of the roots of x² − 8x + 15 = 0? (a) 15 (b) 16 (c) 8 (d) 14
16. 16.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 1 and 16 (c) 2 and 36 (d) 1 and 72
17. 17.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 73 (c) 72 (d) 17
18. 18.What are the roots of x² − 6x + 8 = 0? (a) 1 and 5 (b) 2 and 5 (c) 1 and 8 (d) 2 and 4
19. 19.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1156 (b) 1166 (c) 676 (d) 916
20. 20.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6084 (b) 6404 (c) 6734 (d) 6724
21. 21.If a + b = 13 and ab = 42, what is (a − b)²? (a) 85 (b) 169 (c) 1 (d) 127
22. 22.What is the product of the roots of x² − 7x + 10 = 0? (a) 7 (b) 9 (c) 11 (d) 10
23. 23.What are the roots of x² − 5x + 6 = 0? (a) 1 and 4 (b) 2 and 4 (c) 2 and 3 (d) 1 and 6
24. 24.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 16 (c) 72 (d) 18
25. 25.If a + b = 6 and ab = 8, what is a² + b²? (a) 28 (b) 52 (c) 36 (d) 20
26. 26.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 1)(x + 10) (c) (x + 5)(x + 6) (d) (x − 5)(x − 6)
27. 27.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 484 (b) 494 (c) 324 (d) 404
28. 28.What is the product of the roots of x² − 11x + 24 = 0? (a) 11 (b) 23 (c) 24 (d) 25
29. 29.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 8) (b) (x + 2)(x + 7) (c) (x + 1)(x + 14) (d) (x − 2)(x − 7)
30. 30.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x + 1)(x + 18) (c) (x + 3)(x + 6) (d) (x − 3)(x − 6)
31. 31.If a + b = 16 and ab = 48, what is a² + b²? (a) 256 (b) 160 (c) 208 (d) 352
32. 32.If a + b = 15 and ab = 50, what is a² + b²? (a) 225 (b) 175 (c) 325 (d) 125
33. 33.If a + b = 12 and ab = 11, what is a² + b²? (a) 166 (b) 144 (c) 122 (d) 133
34. 34.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)
35. 35.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 1024 (c) 784 (d) 896
36. 36.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1235 (b) 1225 (c) 925 (d) 625
37. 37.What is the product of the roots of x² − 8x + 12 = 0? (a) 12 (b) 11 (c) 13 (d) 8
38. 38.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1454 (b) 1764 (c) 1444 (d) 1596
39. 39.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1936 (b) 1296 (c) 1584 (d) 1306
40. 40.What is the sum of the roots of x² − 9x + 20 = 0? (a) 9 (b) 20 (c) 8 (d) 10

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