Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 162 · 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 = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 225 (c) 3 (d) 9
2. 2.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 15 (c) 56 (d) 55
3. 3.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1369 (b) 1849 (c) 1379 (d) 1591
4. 4.If a + b = 14 and ab = 45, what is a² + b²? (a) 106 (b) 151 (c) 196 (d) 286
5. 5.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 56) (c) (x − 7)(x − 8) (d) (x + 1)(x + 14)
6. 6.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1369 (b) 1609 (c) 1849 (d) 1859
7. 7.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 971 (b) 961 (c) 901 (d) 841
8. 8.If a + b = 19 and ab = 88, what is (a − b)²? (a) 9 (b) 361 (c) 3 (d) 185
9. 9.If a + b = 18 and ab = 80, what is (a − b)²? (a) 2 (b) 164 (c) 4 (d) 324
10. 10.If a + b = 17 and ab = 72, what is (a − b)²? (a) 145 (b) 217 (c) 1 (d) 289
11. 11.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3146 (b) 4096 (c) 3584 (d) 3136
12. 12.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1306 (b) 1296 (c) 1584 (d) 1936
13. 13.What are the roots of x² − 8x + 15 = 0? (a) 1 and 15 (b) 3 and 5 (c) 2 and 6 (d) 1 and 7
14. 14.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 48) (c) (x + 1)(x + 13) (d) (x + 6)(x + 8)
15. 15.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 5)(x + 6) (c) (x + 1)(x + 10) (d) (x − 5)(x − 6)
16. 16.If a + b = 11 and ab = 18, what is (a − b)²? (a) 121 (b) 85 (c) 7 (d) 49
17. 17.If a + b = 11 and ab = 10, what is a² + b²? (a) 111 (b) 121 (c) 101 (d) 141
18. 18.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6734 (b) 6724 (c) 6084 (d) 6404
19. 19.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9035 (b) 11025 (c) 9975 (d) 9025
20. 20.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 1)(x + 72) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)
21. 21.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8659 (b) 8649 (c) 7569 (d) 8109
22. 22.What is the sum of the roots of x² − 9x + 14 = 0? (a) 8 (b) 9 (c) 14 (d) 10
23. 23.If a + b = 7 and ab = 6, what is (a − b)²? (a) 37 (b) 49 (c) 5 (d) 25
24. 24.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 909 (b) 1099 (c) 1089 (d) 729
25. 25.If a + b = 16 and ab = 48, what is (a − b)²? (a) 256 (b) 64 (c) 8 (d) 160
26. 26.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x + 2)(x + 9) (c) (x + 1)(x + 18) (d) (x − 2)(x − 9)
27. 27.If a + b = 13 and ab = 40, what is (a − b)²? (a) 89 (b) 9 (c) 169 (d) 3
28. 28.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1521 (b) 1691 (c) 1681 (d) 1601
29. 29.What is the sum of the roots of x² − 11x + 18 = 0? (a) 12 (b) 10 (c) 11 (d) 18
30. 30.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 62 (c) 63 (d) 16
31. 31.If a + b = 13 and ab = 36, what is (a − b)²? (a) 97 (b) 5 (c) 169 (d) 25
32. 32.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 16 (d) 18
33. 33.If a + b = 4 and ab = 3, what is (a − b)²? (a) 4 (b) 2 (c) 10 (d) 16
34. 34.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 15) (c) (x − 7)(x − 9) (d) (x + 1)(x + 63)
35. 35.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 16 (c) 15 (d) 14
36. 36.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x − 2)(x − 6) (c) (x + 1)(x + 7) (d) (x + 2)(x + 6)
37. 37.If a + b = 9 and ab = 20, what is (a − b)²? (a) 1 (b) 61 (c) 81 (d) 41
38. 38.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x − 3)(x − 6) (c) (x + 1)(x + 18) (d) (x + 3)(x + 6)
39. 39.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)
40. 40.If a + b = 14 and ab = 48, what is a² + b²? (a) 196 (b) 148 (c) 292 (d) 100

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