Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 595 · 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 61²? (a) 3601 (b) 3721 (c) 3731 (d) 3481
2. 2.What is the sum of the roots of x² − 8x + 15 = 0? (a) 7 (b) 15 (c) 8 (d) 9
3. 3.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3364 (b) 3854 (c) 3604 (d) 3844
4. 4.If a + b = 15 and ab = 50, what is (a − b)²? (a) 125 (b) 25 (c) 5 (d) 225
5. 5.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x + 1)(x + 27) (c) (x + 3)(x + 9) (d) (x − 3)(x − 9)
6. 6.Which is the correct factorisation of x² + 10x + 21? (a) (x + 3)(x + 7) (b) (x + 1)(x + 21) (c) (x − 3)(x − 7) (d) (x + 1)(x + 9)
7. 7.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 396 (b) 484 (c) 334 (d) 324
8. 8.If a + b = 14 and ab = 33, what is (a − b)²? (a) 196 (b) 8 (c) 64 (d) 130
9. 9.If a + b = 6 and ab = 8, what is a² + b²? (a) 28 (b) 52 (c) 20 (d) 36
10. 10.If a + b = 12 and ab = 35, what is (a − b)²? (a) 74 (b) 4 (c) 2 (d) 144
11. 11.What is the product of the roots of x² − 11x + 28 = 0? (a) 28 (b) 11 (c) 27 (d) 29
12. 12.If a + b = 3 and ab = 2, what is (a − b)²? (a) 1 (b) 5 (c) 7 (d) 9
13. 13.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7066 (b) 6416 (c) 7056 (d) 5776
14. 14.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x + 1)(x + 12) (c) (x + 1)(x + 42) (d) (x − 6)(x − 7)
15. 15.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 851 (b) 961 (c) 841 (d) 899
16. 16.If a + b = 9 and ab = 8, what is a² + b²? (a) 65 (b) 97 (c) 81 (d) 73
17. 17.What are the roots of x² − 15x + 56 = 0? (a) 7 and 8 (b) 1 and 56 (c) 2 and 28 (d) 1 and 14
18. 18.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x + 1)(x + 8) (c) (x − 2)(x − 7) (d) (x + 2)(x + 7)
19. 19.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 5486 (b) 4916 (c) 4356 (d) 5476
20. 20.What is the sum of the roots of x² − 9x + 18 = 0? (a) 18 (b) 8 (c) 9 (d) 10
21. 21.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1936 (b) 1584 (c) 1306 (d) 1296
22. 22.If a + b = 10 and ab = 16, what is (a − b)²? (a) 100 (b) 6 (c) 36 (d) 68
23. 23.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 784 (c) 1024 (d) 896
24. 24.If a + b = 15 and ab = 56, what is (a − b)²? (a) 169 (b) 113 (c) 1 (d) 225
25. 25.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 18 (d) 16
26. 26.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8649 (b) 8659 (c) 7569 (d) 8109
27. 27.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)
28. 28.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)
29. 29.If a + b = 21 and ab = 108, what is a² + b²? (a) 657 (b) 333 (c) 441 (d) 225
30. 30.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 72 (c) 73 (d) 17
31. 31.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x + 1)(x + 8) (c) (x + 2)(x + 4) (d) (x − 2)(x − 4)
32. 32.If a + b = 17 and ab = 70, what is (a − b)²? (a) 3 (b) 9 (c) 149 (d) 289
33. 33.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9991 (b) 9419 (c) 9409 (d) 10609
34. 34.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x + 3)(x + 4) (c) (x − 3)(x − 4) (d) (x + 1)(x + 12)
35. 35.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 14) (c) (x + 1)(x + 56) (d) (x + 7)(x + 8)
36. 36.What is the product of the roots of x² − 7x + 12 = 0? (a) 12 (b) 13 (c) 11 (d) 7
37. 37.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x − 4)(x − 6) (c) (x + 1)(x + 9) (d) (x + 4)(x + 6)
38. 38.If a + b = 18 and ab = 72, what is (a − b)²? (a) 36 (b) 180 (c) 324 (d) 6
39. 39.What is the sum of the roots of x² − 15x + 54 = 0? (a) 15 (b) 16 (c) 54 (d) 14
40. 40.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4225 (b) 5635 (c) 4925 (d) 5625

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