Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 255 · 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 95²? (a) 7225 (b) 9035 (c) 8125 (d) 9025
2. 2.What is the sum of the roots of x² − 12x + 32 = 0? (a) 13 (b) 12 (c) 32 (d) 11
3. 3.What is the product of the roots of x² − 9x + 14 = 0? (a) 9 (b) 15 (c) 14 (d) 13
4. 4.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 18) (b) (x + 1)(x + 10) (c) (x + 2)(x + 9) (d) (x − 2)(x − 9)
5. 5.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 635 (b) 225 (c) 625 (d) 425
6. 6.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 361 (b) 401 (c) 451 (d) 441
7. 7.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1849 (b) 1859 (c) 1609 (d) 1369
8. 8.If a + b = 7 and ab = 12, what is (a − b)²? (a) 25 (b) 49 (c) 1 (d) 37
9. 9.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x − 8)(x − 9) (c) (x + 8)(x + 9) (d) (x + 1)(x + 16)
10. 10.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2704 (b) 2304 (c) 2496 (d) 2314
11. 11.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)
12. 12.If a + b = 7 and ab = 6, what is a² + b²? (a) 49 (b) 43 (c) 37 (d) 61
13. 13.What is the sum of the roots of x² − 13x + 42 = 0? (a) 13 (b) 42 (c) 12 (d) 14
14. 14.If a + b = 4 and ab = 3, what is (a − b)²? (a) 2 (b) 4 (c) 10 (d) 16
15. 15.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7056 (b) 6416 (c) 5776 (d) 7066
16. 16.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x + 1)(x + 10) (c) (x − 3)(x − 8) (d) (x + 3)(x + 8)
17. 17.If a + b = 15 and ab = 50, what is a² + b²? (a) 175 (b) 225 (c) 325 (d) 125
18. 18.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 217 (c) 289 (d) 145
19. 19.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 7396 (b) 7406 (c) 8836 (d) 8084
20. 20.If a + b = 12 and ab = 27, what is a² + b²? (a) 90 (b) 117 (c) 198 (d) 144
21. 21.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 9 (c) 3 (d) 441
22. 22.If a + b = 20 and ab = 96, what is a² + b²? (a) 208 (b) 400 (c) 304 (d) 592
23. 23.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 896 (c) 1024 (d) 784
24. 24.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 72 (c) 18 (d) 16
25. 25.What is the product of the roots of x² − 13x + 40 = 0? (a) 40 (b) 13 (c) 41 (d) 39
26. 26.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 10201 (b) 9811 (c) 9801 (d) 9999
27. 27.What are the roots of x² − 14x + 45 = 0? (a) 1 and 45 (b) 3 and 15 (c) 5 and 9 (d) 1 and 13
28. 28.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 400 (c) 202 (d) 4
29. 29.If a + b = 18 and ab = 80, what is (a − b)²? (a) 4 (b) 2 (c) 164 (d) 324
30. 30.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x + 5)(x + 9) (c) (x − 5)(x − 9) (d) (x + 1)(x + 13)
31. 31.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x − 7)(x − 9) (c) (x + 7)(x + 9) (d) (x + 1)(x + 15)
32. 32.If a + b = 11 and ab = 10, what is (a − b)²? (a) 101 (b) 121 (c) 81 (d) 9
33. 33.If a + b = 14 and ab = 48, what is a² + b²? (a) 196 (b) 292 (c) 148 (d) 100
34. 34.What are the roots of x² − 15x + 56 = 0? (a) 2 and 28 (b) 1 and 56 (c) 1 and 14 (d) 7 and 8
35. 35.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1034 (b) 904 (c) 1024 (d) 784
36. 36.What is the sum of the roots of x² − 15x + 54 = 0? (a) 54 (b) 16 (c) 15 (d) 14
37. 37.What are the roots of x² − 12x + 27 = 0? (a) 3 and 9 (b) 1 and 27 (c) 1 and 11 (d) 2 and 10
38. 38.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1521 (b) 1691 (c) 1681 (d) 1601
39. 39.If a + b = 19 and ab = 88, what is a² + b²? (a) 273 (b) 361 (c) 537 (d) 185
40. 40.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 9604 (b) 10404 (c) 10414 (d) 10004

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