Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 551 · 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 are the roots of x² − 16x + 63 = 0? (a) 1 and 15 (b) 3 and 21 (c) 7 and 9 (d) 1 and 63
2. 2.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x + 1)(x + 45) (c) (x + 5)(x + 9) (d) (x − 5)(x − 9)
3. 3.If a + b = 15 and ab = 50, what is a² + b²? (a) 175 (b) 125 (c) 325 (d) 225
4. 4.If a + b = 18 and ab = 80, what is a² + b²? (a) 244 (b) 484 (c) 164 (d) 324
5. 5.If a + b = 5 and ab = 4, what is (a − b)²? (a) 9 (b) 25 (c) 3 (d) 17
6. 6.What are the roots of x² − 14x + 45 = 0? (a) 1 and 45 (b) 5 and 9 (c) 3 and 15 (d) 1 and 13
7. 7.What is the product of the roots of x² − 11x + 30 = 0? (a) 11 (b) 29 (c) 31 (d) 30
8. 8.If a + b = 15 and ab = 54, what is a² + b²? (a) 225 (b) 171 (c) 333 (d) 117
9. 9.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 5194 (b) 5184 (c) 4624 (d) 4904
10. 10.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x − 2)(x − 4) (c) (x + 2)(x + 4) (d) (x + 1)(x + 8)
11. 11.What is the sum of the roots of x² − 9x + 20 = 0? (a) 20 (b) 9 (c) 8 (d) 10
12. 12.Which is the correct factorisation of x² + 8x + 15? (a) (x + 3)(x + 5) (b) (x + 1)(x + 7) (c) (x − 3)(x − 5) (d) (x + 1)(x + 15)
13. 13.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1601 (b) 1691 (c) 1681 (d) 1521
14. 14.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 1)(x + 30) (c) (x − 5)(x − 6) (d) (x + 5)(x + 6)
15. 15.If a + b = 11 and ab = 30, what is (a − b)²? (a) 61 (b) 121 (c) 1 (d) 91
16. 16.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 63) (c) (x + 1)(x + 15) (d) (x − 7)(x − 9)
17. 17.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x + 1)(x + 9) (c) (x − 3)(x − 7) (d) (x + 3)(x + 7)
18. 18.If a + b = 17 and ab = 60, what is (a − b)²? (a) 169 (b) 49 (c) 289 (d) 7
19. 19.What are the roots of x² − 5x + 6 = 0? (a) 1 and 6 (b) 2 and 4 (c) 2 and 3 (d) 1 and 4
20. 20.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8096 (b) 8464 (c) 7754 (d) 7744
21. 21.What are the roots of x² − 14x + 48 = 0? (a) 1 and 48 (b) 2 and 24 (c) 6 and 8 (d) 1 and 13
22. 22.If a + b = 14 and ab = 45, what is a² + b²? (a) 106 (b) 286 (c) 151 (d) 196
23. 23.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6084 (b) 6724 (c) 6396 (d) 6094
24. 24.If a + b = 9 and ab = 20, what is (a − b)²? (a) 1 (b) 61 (c) 81 (d) 41
25. 25.What is the product of the roots of x² − 13x + 40 = 0? (a) 41 (b) 13 (c) 39 (d) 40
26. 26.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)
27. 27.If a + b = 13 and ab = 42, what is a² + b²? (a) 127 (b) 253 (c) 85 (d) 169
28. 28.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 72 (c) 18 (d) 16
29. 29.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8846 (b) 7396 (c) 8836 (d) 8116
30. 30.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 2 and 36 (c) 8 and 9 (d) 1 and 72
31. 31.What are the roots of x² − 9x + 18 = 0? (a) 1 and 18 (b) 2 and 9 (c) 3 and 6 (d) 1 and 8
32. 32.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4761 (b) 5041 (c) 4771 (d) 4899
33. 33.What are the roots of x² − 13x + 40 = 0? (a) 2 and 20 (b) 5 and 8 (c) 1 and 12 (d) 1 and 40
34. 34.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6889 (b) 6391 (c) 5929 (d) 5939
35. 35.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x − 7)(x − 8) (c) (x + 7)(x + 8) (d) (x + 1)(x + 56)
36. 36.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 5786 (b) 6384 (c) 7056 (d) 5776
37. 37.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 1024 (c) 784 (d) 896
38. 38.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x + 1)(x + 13) (c) (x − 6)(x − 8) (d) (x + 6)(x + 8)
39. 39.If a + b = 11 and ab = 24, what is a² + b²? (a) 97 (b) 73 (c) 121 (d) 169
40. 40.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x + 1)(x + 8) (c) (x + 2)(x + 7) (d) (x − 2)(x − 7)

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