Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 712 · 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 + 1)(x + 7) (b) (x + 3)(x + 5) (c) (x + 1)(x + 15) (d) (x − 3)(x − 5)
2. 2.What are the roots of x² − 6x + 8 = 0? (a) 2 and 5 (b) 1 and 5 (c) 1 and 8 (d) 2 and 4
3. 3.If a + b = 11 and ab = 30, what is a² + b²? (a) 61 (b) 91 (c) 181 (d) 121
4. 4.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1936 (b) 1306 (c) 1296 (d) 1584
5. 5.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4235 (b) 4875 (c) 5625 (d) 4225
6. 6.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 1)(x + 14) (c) (x + 6)(x + 9) (d) (x − 6)(x − 9)
7. 7.If a + b = 9 and ab = 20, what is a² + b²? (a) 61 (b) 121 (c) 41 (d) 81
8. 8.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 1)(x + 36) (c) (x + 1)(x + 12) (d) (x + 4)(x + 9)
9. 9.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 9025 (b) 11025 (c) 11035 (d) 10025
10. 10.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 399 (b) 441 (c) 371 (d) 361
11. 11.If a + b = 10 and ab = 16, what is a² + b²? (a) 132 (b) 84 (c) 68 (d) 100
12. 12.If a + b = 16 and ab = 48, what is (a − b)²? (a) 8 (b) 64 (c) 160 (d) 256
13. 13.If a + b = 14 and ab = 45, what is (a − b)²? (a) 196 (b) 106 (c) 16 (d) 4
14. 14.Which is the correct factorisation of x² + 10x + 24? (a) (x + 4)(x + 6) (b) (x + 1)(x + 9) (c) (x + 1)(x + 24) (d) (x − 4)(x − 6)
15. 15.What is the product of the roots of x² − 9x + 14 = 0? (a) 13 (b) 9 (c) 15 (d) 14
16. 16.What is the sum of the roots of x² − 13x + 42 = 0? (a) 14 (b) 42 (c) 12 (d) 13
17. 17.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1764 (b) 1444 (c) 1596 (d) 1454
18. 18.If a + b = 19 and ab = 88, what is (a − b)²? (a) 3 (b) 185 (c) 361 (d) 9
19. 19.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2484 (b) 2126 (c) 2916 (d) 2116
20. 20.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 17 (c) 16 (d) 63
21. 21.Which is the correct factorisation of x² + 7x + 12? (a) (x + 3)(x + 4) (b) (x − 3)(x − 4) (c) (x + 1)(x + 6) (d) (x + 1)(x + 12)
22. 22.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 891 (b) 739 (c) 1089 (d) 729
23. 23.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x + 3)(x + 6) (c) (x − 3)(x − 6) (d) (x + 1)(x + 8)
24. 24.If a + b = 12 and ab = 20, what is (a − b)²? (a) 144 (b) 64 (c) 104 (d) 8
25. 25.Which is the correct factorisation of x² + 12x + 27? (a) (x + 3)(x + 9) (b) (x + 1)(x + 27) (c) (x − 3)(x − 9) (d) (x + 1)(x + 11)
26. 26.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)
27. 27.If a + b = 3 and ab = 2, what is a² + b²? (a) 9 (b) 7 (c) 5 (d) 13
28. 28.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 541 (c) 361 (d) 181
29. 29.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 1 and 72 (c) 2 and 36 (d) 8 and 9
30. 30.If a + b = 16 and ab = 63, what is (a − b)²? (a) 2 (b) 256 (c) 130 (d) 4
31. 31.If a + b = 18 and ab = 80, what is (a − b)²? (a) 324 (b) 164 (c) 2 (d) 4
32. 32.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1764 (b) 1604 (c) 1774 (d) 1444
33. 33.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x − 6)(x − 8) (c) (x + 6)(x + 8) (d) (x + 1)(x + 13)
34. 34.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2926 (b) 2116 (c) 2516 (d) 2916
35. 35.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x − 2)(x − 4) (c) (x + 1)(x + 5) (d) (x + 2)(x + 4)
36. 36.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 55 (c) 56 (d) 57
37. 37.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 334 (b) 396 (c) 484 (d) 324
38. 38.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3584 (b) 3136 (c) 3146 (d) 4096
39. 39.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 17 (c) 72 (d) 73
40. 40.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 416 (c) 586 (d) 576

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