Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 449 · 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 − 9x + 18 = 0? (a) 1 and 8 (b) 2 and 9 (c) 3 and 6 (d) 1 and 18
2. 2.If a + b = 9 and ab = 8, what is a² + b²? (a) 81 (b) 97 (c) 73 (d) 65
3. 3.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 8 and 9 (c) 2 and 36 (d) 1 and 72
4. 4.If a + b = 12 and ab = 20, what is (a − b)²? (a) 64 (b) 8 (c) 144 (d) 104
5. 5.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1625 (b) 2035 (c) 2025 (d) 1225
6. 6.If a + b = 13 and ab = 40, what is a² + b²? (a) 129 (b) 169 (c) 249 (d) 89
7. 7.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 55 (c) 56 (d) 57
8. 8.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 5939 (b) 5929 (c) 6391 (d) 6889
9. 9.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6561 (b) 6241 (c) 6571 (d) 6401
10. 10.If a + b = 12 and ab = 35, what is (a − b)²? (a) 144 (b) 74 (c) 2 (d) 4
11. 11.If a + b = 15 and ab = 44, what is a² + b²? (a) 225 (b) 181 (c) 137 (d) 313
12. 12.What are the roots of x² − 15x + 56 = 0? (a) 7 and 8 (b) 2 and 28 (c) 1 and 56 (d) 1 and 14
13. 13.What is the sum of the roots of x² − 13x + 42 = 0? (a) 12 (b) 42 (c) 13 (d) 14
14. 14.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x + 1)(x + 6) (c) (x − 2)(x − 3) (d) (x + 2)(x + 3)
15. 15.What is the sum of the roots of x² − 14x + 45 = 0? (a) 13 (b) 45 (c) 14 (d) 15
16. 16.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
17. 17.What is the sum of the roots of x² − 11x + 30 = 0? (a) 30 (b) 11 (c) 10 (d) 12
18. 18.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)
19. 19.If a + b = 9 and ab = 20, what is (a − b)²? (a) 81 (b) 41 (c) 61 (d) 1
20. 20.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 4)(x + 6) (c) (x + 1)(x + 24) (d) (x + 1)(x + 9)
21. 21.What is the product of the roots of x² − 12x + 32 = 0? (a) 31 (b) 32 (c) 33 (d) 12
22. 22.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 441 (b) 451 (c) 401 (d) 361
23. 23.What is the product of the roots of x² − 11x + 24 = 0? (a) 11 (b) 24 (c) 25 (d) 23
24. 24.If a + b = 13 and ab = 42, what is (a − b)²? (a) 85 (b) 169 (c) 127 (d) 1
25. 25.What is the product of the roots of x² − 12x + 35 = 0? (a) 35 (b) 36 (c) 34 (d) 12
26. 26.If a + b = 18 and ab = 80, what is a² + b²? (a) 484 (b) 164 (c) 244 (d) 324
27. 27.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 7396 (b) 8084 (c) 7406 (d) 8836
28. 28.If a + b = 17 and ab = 66, what is a² + b²? (a) 223 (b) 289 (c) 157 (d) 421
29. 29.If a + b = 17 and ab = 70, what is a² + b²? (a) 149 (b) 219 (c) 429 (d) 289
30. 30.What is the sum of the roots of x² − 12x + 32 = 0? (a) 12 (b) 13 (c) 32 (d) 11
31. 31.What is the sum of the roots of x² − 5x + 6 = 0? (a) 7 (b) 5 (c) 4 (d) 6
32. 32.If a + b = 12 and ab = 35, what is a² + b²? (a) 74 (b) 144 (c) 214 (d) 109
33. 33.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 925 (b) 625 (c) 1235 (d) 1225
34. 34.Which is the correct factorisation of x² + 11x + 28? (a) (x + 4)(x + 7) (b) (x − 4)(x − 7) (c) (x + 1)(x + 10) (d) (x + 1)(x + 28)
35. 35.If a + b = 18 and ab = 80, what is (a − b)²? (a) 164 (b) 2 (c) 324 (d) 4
36. 36.If a + b = 15 and ab = 56, what is a² + b²? (a) 169 (b) 337 (c) 113 (d) 225
37. 37.If a + b = 5 and ab = 6, what is a² + b²? (a) 25 (b) 19 (c) 13 (d) 37
38. 38.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 625 (b) 225 (c) 375 (d) 235
39. 39.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 8125 (b) 7225 (c) 9025 (d) 9035
40. 40.What is the sum of the roots of x² − 12x + 27 = 0? (a) 11 (b) 13 (c) 27 (d) 12

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