Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 411 · 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 + 1)(x + 48) (b) (x − 6)(x − 8) (c) (x + 1)(x + 13) (d) (x + 6)(x + 8)
2. 2.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 72)
3. 3.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7754 (b) 8096 (c) 8464 (d) 7744
4. 4.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1531 (b) 1681 (c) 1599 (d) 1521
5. 5.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 17 (d) 16
6. 6.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)
7. 7.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4235 (b) 4875 (c) 4225 (d) 5625
8. 8.If a + b = 10 and ab = 21, what is a² + b²? (a) 142 (b) 100 (c) 58 (d) 79
9. 9.What are the roots of x² − 12x + 27 = 0? (a) 1 and 27 (b) 3 and 9 (c) 1 and 11 (d) 2 and 10
10. 10.If a + b = 7 and ab = 10, what is (a − b)²? (a) 29 (b) 9 (c) 3 (d) 49
11. 11.If a + b = 13 and ab = 30, what is a² + b²? (a) 169 (b) 139 (c) 109 (d) 229
12. 12.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1379 (b) 1849 (c) 1369 (d) 1591
13. 13.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 729 (b) 909 (c) 1099 (d) 1089
14. 14.What are the roots of x² − 13x + 42 = 0? (a) 2 and 21 (b) 1 and 12 (c) 6 and 7 (d) 1 and 42
15. 15.If a + b = 21 and ab = 108, what is (a − b)²? (a) 3 (b) 441 (c) 225 (d) 9
16. 16.What is the sum of the roots of x² − 14x + 45 = 0? (a) 14 (b) 15 (c) 45 (d) 13
17. 17.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)
18. 18.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3601 (b) 3731 (c) 3721 (d) 3481
19. 19.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 15 (c) 55 (d) 56
20. 20.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 625 (b) 1225 (c) 925 (d) 1235
21. 21.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x − 3)(x − 4) (c) (x + 3)(x + 4) (d) (x + 1)(x + 6)
22. 22.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 9604 (b) 10404 (c) 10414 (d) 10004
23. 23.If a + b = 16 and ab = 55, what is (a − b)²? (a) 256 (b) 6 (c) 36 (d) 146
24. 24.What is the product of the roots of x² − 14x + 45 = 0? (a) 14 (b) 46 (c) 45 (d) 44
25. 25.What is the sum of the roots of x² − 11x + 30 = 0? (a) 10 (b) 11 (c) 12 (d) 30
26. 26.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x + 4)(x + 8) (c) (x + 1)(x + 11) (d) (x − 4)(x − 8)
27. 27.If a + b = 12 and ab = 32, what is a² + b²? (a) 112 (b) 144 (c) 208 (d) 80
28. 28.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)
29. 29.What is the product of the roots of x² − 6x + 8 = 0? (a) 6 (b) 8 (c) 7 (d) 9
30. 30.Which is the correct factorisation of x² + 10x + 21? (a) (x + 3)(x + 7) (b) (x − 3)(x − 7) (c) (x + 1)(x + 21) (d) (x + 1)(x + 9)
31. 31.What is the product of the roots of x² − 15x + 54 = 0? (a) 54 (b) 55 (c) 15 (d) 53
32. 32.What is the product of the roots of x² − 16x + 63 = 0? (a) 63 (b) 16 (c) 64 (d) 62
33. 33.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1596 (b) 1444 (c) 1454 (d) 1764
34. 34.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1584 (b) 1936 (c) 1296 (d) 1306
35. 35.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 1)(x + 20) (c) (x + 4)(x + 5) (d) (x + 1)(x + 8)
36. 36.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 1)(x + 10) (c) (x + 5)(x + 6) (d) (x − 5)(x − 6)
37. 37.What is the product of the roots of x² − 14x + 48 = 0? (a) 49 (b) 47 (c) 14 (d) 48
38. 38.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 529 (b) 391 (c) 289 (d) 299
39. 39.Which is the correct factorisation of x² + 13x + 36? (a) (x + 4)(x + 9) (b) (x + 1)(x + 12) (c) (x + 1)(x + 36) (d) (x − 4)(x − 9)
40. 40.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 7396 (b) 8084 (c) 7406 (d) 8836

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