Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 548 · 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 = 11 and ab = 24, what is (a − b)²? (a) 5 (b) 25 (c) 121 (d) 73
2. 2.If a + b = 7 and ab = 6, what is (a − b)²? (a) 5 (b) 25 (c) 49 (d) 37
3. 3.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 225 (b) 425 (c) 625 (d) 635
4. 4.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
5. 5.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 1)(x + 11) (c) (x + 1)(x + 35) (d) (x + 5)(x + 7)
6. 6.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 17 (c) 71 (d) 72
7. 7.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 9025 (b) 7225 (c) 8075 (d) 7235
8. 8.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2411 (b) 2499 (c) 2601 (d) 2401
9. 9.If a + b = 15 and ab = 50, what is a² + b²? (a) 325 (b) 125 (c) 175 (d) 225
10. 10.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 14) (c) (x + 7)(x + 8) (d) (x + 1)(x + 56)
11. 11.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1034 (b) 1024 (c) 904 (d) 784
12. 12.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x + 3)(x + 6) (c) (x + 1)(x + 8) (d) (x − 3)(x − 6)
13. 13.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 909 (b) 1089 (c) 729 (d) 1099
14. 14.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10201 (b) 10211 (c) 10001 (d) 9801
15. 15.If a + b = 18 and ab = 77, what is (a − b)²? (a) 16 (b) 324 (c) 4 (d) 170
16. 16.If a + b = 18 and ab = 80, what is a² + b²? (a) 484 (b) 324 (c) 244 (d) 164
17. 17.What are the roots of x² − 10x + 24 = 0? (a) 2 and 12 (b) 1 and 9 (c) 4 and 6 (d) 1 and 24
18. 18.What are the roots of x² − 10x + 16 = 0? (a) 3 and 7 (b) 1 and 16 (c) 2 and 8 (d) 1 and 9
19. 19.If a + b = 15 and ab = 54, what is a² + b²? (a) 171 (b) 333 (c) 117 (d) 225
20. 20.If a + b = 20 and ab = 99, what is (a − b)²? (a) 4 (b) 400 (c) 2 (d) 202
21. 21.If a + b = 9 and ab = 14, what is (a − b)²? (a) 53 (b) 25 (c) 81 (d) 5
22. 22.What is the product of the roots of x² − 14x + 45 = 0? (a) 44 (b) 14 (c) 46 (d) 45
23. 23.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)
24. 24.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x − 3)(x − 4) (c) (x + 1)(x + 6) (d) (x + 3)(x + 4)
25. 25.If a + b = 7 and ab = 10, what is (a − b)²? (a) 29 (b) 9 (c) 49 (d) 3
26. 26.What are the roots of x² − 8x + 15 = 0? (a) 1 and 15 (b) 2 and 6 (c) 1 and 7 (d) 3 and 5
27. 27.If a + b = 16 and ab = 55, what is a² + b²? (a) 146 (b) 201 (c) 256 (d) 366
28. 28.What is the sum of the roots of x² − 14x + 45 = 0? (a) 15 (b) 13 (c) 14 (d) 45
29. 29.What are the roots of x² − 13x + 40 = 0? (a) 5 and 8 (b) 1 and 12 (c) 2 and 20 (d) 1 and 40
30. 30.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4761 (b) 4771 (c) 4899 (d) 5041
31. 31.What are the roots of x² − 11x + 30 = 0? (a) 1 and 30 (b) 1 and 10 (c) 5 and 6 (d) 2 and 15
32. 32.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 334 (b) 324 (c) 396 (d) 484
33. 33.If a + b = 11 and ab = 10, what is (a − b)²? (a) 121 (b) 81 (c) 9 (d) 101
34. 34.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x + 1)(x + 40) (c) (x − 5)(x − 8) (d) (x + 5)(x + 8)
35. 35.If a + b = 11 and ab = 28, what is (a − b)²? (a) 3 (b) 65 (c) 9 (d) 121
36. 36.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 625 (b) 375 (c) 235 (d) 225
37. 37.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4499 (b) 4489 (c) 5329 (d) 4891
38. 38.If a + b = 13 and ab = 12, what is (a − b)²? (a) 145 (b) 11 (c) 169 (d) 121
39. 39.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6889 (b) 5939 (c) 6391 (d) 5929
40. 40.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 6416 (b) 5776 (c) 7066 (d) 7056

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