Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 43 · 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.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8281 (b) 8291 (c) 8101 (d) 7921
2. 2.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 784 (c) 1024 (d) 896
3. 3.What is the product of the roots of x² − 11x + 18 = 0? (a) 11 (b) 18 (c) 19 (d) 17
4. 4.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 361 (b) 401 (c) 451 (d) 441
5. 5.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x − 4)(x − 5) (c) (x + 4)(x + 5) (d) (x + 1)(x + 8)
6. 6.If a + b = 18 and ab = 77, what is (a − b)²? (a) 324 (b) 4 (c) 16 (d) 170
7. 7.If a + b = 13 and ab = 22, what is (a − b)²? (a) 9 (b) 81 (c) 125 (d) 169
8. 8.What is the product of the roots of x² − 13x + 40 = 0? (a) 41 (b) 39 (c) 40 (d) 13
9. 9.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 6)(x + 8) (c) (x + 1)(x + 48) (d) (x − 6)(x − 8)
10. 10.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
11. 11.If a + b = 11 and ab = 18, what is a² + b²? (a) 157 (b) 121 (c) 103 (d) 85
12. 12.What is the product of the roots of x² − 7x + 10 = 0? (a) 9 (b) 10 (c) 11 (d) 7
13. 13.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x − 4)(x − 7) (c) (x + 1)(x + 10) (d) (x + 4)(x + 7)
14. 14.If a + b = 7 and ab = 12, what is a² + b²? (a) 25 (b) 73 (c) 49 (d) 37
15. 15.What are the roots of x² − 15x + 56 = 0? (a) 2 and 28 (b) 7 and 8 (c) 1 and 56 (d) 1 and 14
16. 16.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x + 1)(x + 15) (c) (x − 3)(x − 5) (d) (x + 3)(x + 5)
17. 17.If a + b = 19 and ab = 88, what is (a − b)²? (a) 3 (b) 361 (c) 9 (d) 185
18. 18.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 2025 (b) 1225 (c) 1575 (d) 1235
19. 19.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9999 (b) 10201 (c) 9811 (d) 9801
20. 20.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 1)(x + 72)
21. 21.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)
22. 22.If a + b = 13 and ab = 36, what is a² + b²? (a) 97 (b) 241 (c) 133 (d) 169
23. 23.What is the sum of the roots of x² − 12x + 27 = 0? (a) 27 (b) 11 (c) 13 (d) 12
24. 24.What are the roots of x² − 7x + 10 = 0? (a) 1 and 10 (b) 1 and 6 (c) 3 and 4 (d) 2 and 5
25. 25.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 16 (c) 14 (d) 56
26. 26.What are the roots of x² − 16x + 63 = 0? (a) 1 and 15 (b) 7 and 9 (c) 1 and 63 (d) 3 and 21
27. 27.What are the roots of x² − 14x + 48 = 0? (a) 1 and 13 (b) 2 and 24 (c) 1 and 48 (d) 6 and 8
28. 28.What are the roots of x² − 9x + 18 = 0? (a) 3 and 6 (b) 1 and 8 (c) 1 and 18 (d) 2 and 9
29. 29.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 9025 (b) 7225 (c) 9035 (d) 8125
30. 30.What is the product of the roots of x² − 16x + 63 = 0? (a) 16 (b) 63 (c) 64 (d) 62
31. 31.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 901 (b) 971 (c) 961 (d) 841
32. 32.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 625 (b) 925 (c) 1235 (d) 1225
33. 33.If a + b = 16 and ab = 63, what is a² + b²? (a) 382 (b) 130 (c) 193 (d) 256
34. 34.If a + b = 11 and ab = 24, what is (a − b)²? (a) 5 (b) 121 (c) 73 (d) 25
35. 35.What are the roots of x² − 10x + 24 = 0? (a) 1 and 9 (b) 1 and 24 (c) 2 and 12 (d) 4 and 6
36. 36.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x − 6)(x − 7) (c) (x + 1)(x + 42) (d) (x + 6)(x + 7)
37. 37.If a + b = 14 and ab = 33, what is a² + b²? (a) 196 (b) 163 (c) 130 (d) 262
38. 38.If a + b = 10 and ab = 16, what is a² + b²? (a) 84 (b) 68 (c) 100 (d) 132
39. 39.What is the sum of the roots of x² − 10x + 24 = 0? (a) 10 (b) 9 (c) 24 (d) 11
40. 40.If a + b = 3 and ab = 2, what is (a − b)²? (a) 5 (b) 1 (c) 7 (d) 9

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