Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 258 · 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 + 7x + 12? (a) (x + 3)(x + 4) (b) (x + 1)(x + 6) (c) (x + 1)(x + 12) (d) (x − 3)(x − 4)
2. 2.What is the product of the roots of x² − 13x + 36 = 0? (a) 37 (b) 13 (c) 35 (d) 36
3. 3.If a + b = 8 and ab = 15, what is a² + b²? (a) 49 (b) 34 (c) 94 (d) 64
4. 4.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 289 (b) 391 (c) 529 (d) 299
5. 5.What are the roots of x² − 10x + 24 = 0? (a) 1 and 9 (b) 4 and 6 (c) 2 and 12 (d) 1 and 24
6. 6.If a + b = 9 and ab = 20, what is a² + b²? (a) 61 (b) 121 (c) 41 (d) 81
7. 7.What are the roots of x² − 13x + 36 = 0? (a) 1 and 36 (b) 2 and 18 (c) 1 and 12 (d) 4 and 9
8. 8.What are the roots of x² − 9x + 20 = 0? (a) 2 and 10 (b) 4 and 5 (c) 1 and 20 (d) 1 and 8
9. 9.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 56 (c) 15 (d) 57
10. 10.What is the sum of the roots of x² − 14x + 45 = 0? (a) 45 (b) 13 (c) 14 (d) 15
11. 11.If a + b = 6 and ab = 8, what is a² + b²? (a) 20 (b) 52 (c) 28 (d) 36
12. 12.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 5)(x + 9) (c) (x + 1)(x + 13) (d) (x + 1)(x + 45)
13. 13.What are the roots of x² − 11x + 24 = 0? (a) 1 and 10 (b) 3 and 8 (c) 1 and 24 (d) 2 and 12
14. 14.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1616 (b) 1946 (c) 1296 (d) 1936
15. 15.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6889 (b) 6899 (c) 6409 (d) 5929
16. 16.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6251 (b) 6241 (c) 6561 (d) 6399
17. 17.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x + 7)(x + 8) (d) (x − 7)(x − 8)
18. 18.What are the roots of x² − 11x + 28 = 0? (a) 2 and 14 (b) 1 and 10 (c) 1 and 28 (d) 4 and 7
19. 19.If a + b = 13 and ab = 30, what is (a − b)²? (a) 7 (b) 109 (c) 169 (d) 49
20. 20.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x + 6)(x + 9) (c) (x − 6)(x − 9) (d) (x + 1)(x + 54)
21. 21.If a + b = 13 and ab = 36, what is (a − b)²? (a) 5 (b) 25 (c) 97 (d) 169
22. 22.If a + b = 11 and ab = 18, what is a² + b²? (a) 157 (b) 121 (c) 103 (d) 85
23. 23.If a + b = 3 and ab = 2, what is (a − b)²? (a) 5 (b) 9 (c) 7 (d) 1
24. 24.If a + b = 8 and ab = 12, what is a² + b²? (a) 64 (b) 40 (c) 88 (d) 52
25. 25.What are the roots of x² − 5x + 6 = 0? (a) 2 and 4 (b) 1 and 6 (c) 1 and 4 (d) 2 and 3
26. 26.If a + b = 17 and ab = 72, what is a² + b²? (a) 145 (b) 217 (c) 433 (d) 289
27. 27.If a + b = 10 and ab = 24, what is (a − b)²? (a) 100 (b) 2 (c) 4 (d) 52
28. 28.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 909 (b) 1089 (c) 729 (d) 1099
29. 29.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 925 (b) 1235 (c) 1225 (d) 625
30. 30.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x + 1)(x + 16)
31. 31.What are the roots of x² − 16x + 63 = 0? (a) 1 and 63 (b) 3 and 21 (c) 1 and 15 (d) 7 and 9
32. 32.If a + b = 9 and ab = 18, what is a² + b²? (a) 45 (b) 81 (c) 117 (d) 63
33. 33.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 17 (c) 72 (d) 71
34. 34.If a + b = 20 and ab = 96, what is (a − b)²? (a) 4 (b) 208 (c) 400 (d) 16
35. 35.If a + b = 18 and ab = 72, what is a² + b²? (a) 180 (b) 324 (c) 468 (d) 252
36. 36.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 7225 (b) 5635 (c) 6375 (d) 5625
37. 37.If a + b = 19 and ab = 84, what is a² + b²? (a) 193 (b) 529 (c) 277 (d) 361
38. 38.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1774 (b) 1604 (c) 1444 (d) 1764
39. 39.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
40. 40.What is the product of the roots of x² − 11x + 30 = 0? (a) 11 (b) 29 (c) 31 (d) 30

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