Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 538 · 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 + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 6)(x + 8) (c) (x + 1)(x + 48) (d) (x + 1)(x + 13)
2. 2.What is the product of the roots of x² − 10x + 21 = 0? (a) 20 (b) 22 (c) 10 (d) 21
3. 3.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 586 (b) 416 (c) 576 (d) 256
4. 4.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)
5. 5.If a + b = 7 and ab = 12, what is a² + b²? (a) 73 (b) 25 (c) 37 (d) 49
6. 6.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 15 (c) 56 (d) 57
7. 7.What are the roots of x² − 13x + 42 = 0? (a) 1 and 12 (b) 1 and 42 (c) 6 and 7 (d) 2 and 21
8. 8.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
9. 9.If a + b = 11 and ab = 28, what is (a − b)²? (a) 121 (b) 3 (c) 9 (d) 65
10. 10.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 334 (b) 484 (c) 324 (d) 396
11. 11.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 63) (c) (x + 7)(x + 9) (d) (x + 1)(x + 15)
12. 12.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7056 (b) 5776 (c) 7066 (d) 6416
13. 13.If a + b = 13 and ab = 42, what is (a − b)²? (a) 169 (b) 127 (c) 1 (d) 85
14. 14.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 14) (c) (x + 7)(x + 8) (d) (x + 1)(x + 56)
15. 15.If a + b = 11 and ab = 30, what is a² + b²? (a) 181 (b) 121 (c) 91 (d) 61
16. 16.What is the sum of the roots of x² − 9x + 18 = 0? (a) 8 (b) 10 (c) 9 (d) 18
17. 17.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 18) (b) (x − 2)(x − 9) (c) (x + 2)(x + 9) (d) (x + 1)(x + 10)
18. 18.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x + 1)(x + 7) (c) (x + 3)(x + 5) (d) (x − 3)(x − 5)
19. 19.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2714 (b) 2304 (c) 2504 (d) 2704
20. 20.If a + b = 17 and ab = 60, what is a² + b²? (a) 169 (b) 289 (c) 229 (d) 409
21. 21.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6724 (b) 6396 (c) 6084 (d) 6094
22. 22.What are the roots of x² − 12x + 27 = 0? (a) 2 and 10 (b) 1 and 11 (c) 3 and 9 (d) 1 and 27
23. 23.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 891 (b) 1089 (c) 739 (d) 729
24. 24.What is the sum of the roots of x² − 8x + 12 = 0? (a) 12 (b) 8 (c) 9 (d) 7
25. 25.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 5635 (b) 4225 (c) 5625 (d) 4925
26. 26.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1936 (b) 1306 (c) 1296 (d) 1584
27. 27.If a + b = 6 and ab = 5, what is (a − b)²? (a) 4 (b) 36 (c) 26 (d) 16
28. 28.Which is the correct factorisation of x² + 10x + 21? (a) (x + 3)(x + 7) (b) (x + 1)(x + 9) (c) (x − 3)(x − 7) (d) (x + 1)(x + 21)
29. 29.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 18 (c) 72 (d) 17
30. 30.What are the roots of x² − 9x + 20 = 0? (a) 2 and 10 (b) 1 and 8 (c) 4 and 5 (d) 1 and 20
31. 31.What are the roots of x² − 10x + 16 = 0? (a) 3 and 7 (b) 2 and 8 (c) 1 and 9 (d) 1 and 16
32. 32.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x + 1)(x + 20) (c) (x + 4)(x + 5) (d) (x − 4)(x − 5)
33. 33.What is the product of the roots of x² − 5x + 6 = 0? (a) 8 (b) 7 (c) 5 (d) 6
34. 34.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 9409 (b) 10619 (c) 10009 (d) 10609
35. 35.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x − 6)(x − 7) (c) (x + 1)(x + 12) (d) (x + 6)(x + 7)
36. 36.If a + b = 11 and ab = 18, what is (a − b)²? (a) 49 (b) 7 (c) 121 (d) 85
37. 37.What is the product of the roots of x² − 11x + 18 = 0? (a) 11 (b) 17 (c) 18 (d) 19
38. 38.If a + b = 16 and ab = 63, what is (a − b)²? (a) 256 (b) 130 (c) 4 (d) 2
39. 39.If a + b = 12 and ab = 32, what is a² + b²? (a) 80 (b) 112 (c) 208 (d) 144
40. 40.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 5329 (b) 4489 (c) 4891 (d) 4499

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