Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 819 · 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 + 8x + 15? (a) (x + 3)(x + 5) (b) (x + 1)(x + 15) (c) (x + 1)(x + 7) (d) (x − 3)(x − 5)
2. 2.If a + b = 12 and ab = 27, what is (a − b)²? (a) 6 (b) 90 (c) 36 (d) 144
3. 3.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 9) (b) (x + 1)(x + 24) (c) (x + 4)(x + 6) (d) (x − 4)(x − 6)
4. 4.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6409 (b) 6899 (c) 6889 (d) 5929
5. 5.If a + b = 17 and ab = 70, what is a² + b²? (a) 149 (b) 429 (c) 289 (d) 219
6. 6.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x − 7)(x − 8) (c) (x + 1)(x + 56) (d) (x + 7)(x + 8)
7. 7.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 635 (b) 225 (c) 625 (d) 425
8. 8.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6241 (b) 6399 (c) 6251 (d) 6561
9. 9.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1584 (b) 1936 (c) 1306 (d) 1296
10. 10.If a + b = 19 and ab = 88, what is a² + b²? (a) 361 (b) 537 (c) 185 (d) 273
11. 11.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10009 (b) 9409 (c) 10609 (d) 10619
12. 12.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x + 4)(x + 7) (c) (x − 4)(x − 7) (d) (x + 1)(x + 28)
13. 13.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 5)(x + 8) (c) (x + 1)(x + 40) (d) (x + 1)(x + 12)
14. 14.If a + b = 14 and ab = 24, what is (a − b)²? (a) 196 (b) 10 (c) 148 (d) 100
15. 15.If a + b = 17 and ab = 66, what is a² + b²? (a) 421 (b) 223 (c) 157 (d) 289
16. 16.If a + b = 15 and ab = 36, what is a² + b²? (a) 225 (b) 297 (c) 153 (d) 189
17. 17.If a + b = 15 and ab = 56, what is a² + b²? (a) 169 (b) 337 (c) 225 (d) 113
18. 18.If a + b = 11 and ab = 30, what is a² + b²? (a) 121 (b) 91 (c) 61 (d) 181
19. 19.If a + b = 14 and ab = 40, what is a² + b²? (a) 276 (b) 196 (c) 116 (d) 156
20. 20.What are the roots of x² − 12x + 35 = 0? (a) 1 and 11 (b) 1 and 35 (c) 5 and 7 (d) 2 and 10
21. 21.What is the sum of the roots of x² − 11x + 24 = 0? (a) 10 (b) 24 (c) 11 (d) 12
22. 22.What is the sum of the roots of x² − 6x + 8 = 0? (a) 7 (b) 6 (c) 8 (d) 5
23. 23.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x + 1)(x + 16) (c) (x − 2)(x − 8) (d) (x + 2)(x + 8)
24. 24.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7066 (b) 6416 (c) 7056 (d) 5776
25. 25.If a + b = 13 and ab = 42, what is (a − b)²? (a) 127 (b) 1 (c) 85 (d) 169
26. 26.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1379 (b) 1849 (c) 1369 (d) 1591
27. 27.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 1)(x + 63) (c) (x + 7)(x + 9) (d) (x − 7)(x − 9)
28. 28.What is the sum of the roots of x² − 14x + 48 = 0? (a) 14 (b) 13 (c) 48 (d) 15
29. 29.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2025 (b) 2475 (c) 2035 (d) 3025
30. 30.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4489 (b) 4891 (c) 5329 (d) 4499
31. 31.What is the sum of the roots of x² − 7x + 10 = 0? (a) 7 (b) 10 (c) 6 (d) 8
32. 32.If a + b = 10 and ab = 9, what is (a − b)²? (a) 82 (b) 64 (c) 8 (d) 100
33. 33.If a + b = 8 and ab = 12, what is (a − b)²? (a) 4 (b) 16 (c) 64 (d) 40
34. 34.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 739 (b) 729 (c) 891 (d) 1089
35. 35.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 6) (c) (x + 3)(x + 4) (d) (x + 1)(x + 12)
36. 36.If a + b = 18 and ab = 77, what is (a − b)²? (a) 170 (b) 16 (c) 4 (d) 324
37. 37.What are the roots of x² − 15x + 54 = 0? (a) 1 and 14 (b) 1 and 54 (c) 2 and 27 (d) 6 and 9
38. 38.What is the product of the roots of x² − 14x + 48 = 0? (a) 14 (b) 49 (c) 47 (d) 48
39. 39.If a + b = 8 and ab = 15, what is a² + b²? (a) 64 (b) 94 (c) 49 (d) 34
40. 40.If a + b = 11 and ab = 18, what is (a − b)²? (a) 121 (b) 7 (c) 85 (d) 49

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