Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 909 · 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.If a + b = 13 and ab = 36, what is a² + b²? (a) 241 (b) 133 (c) 169 (d) 97
2. 2.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1584 (b) 1936 (c) 1296 (d) 1306
3. 3.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x − 6)(x − 7) (c) (x + 1)(x + 42) (d) (x + 1)(x + 12)
4. 4.If a + b = 10 and ab = 21, what is a² + b²? (a) 142 (b) 58 (c) 79 (d) 100
5. 5.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9999 (b) 9801 (c) 10201 (d) 9811
6. 6.What are the roots of x² − 13x + 36 = 0? (a) 1 and 12 (b) 1 and 36 (c) 4 and 9 (d) 2 and 18
7. 7.What are the roots of x² − 12x + 27 = 0? (a) 2 and 10 (b) 3 and 9 (c) 1 and 27 (d) 1 and 11
8. 8.If a + b = 9 and ab = 20, what is a² + b²? (a) 81 (b) 41 (c) 121 (d) 61
9. 9.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 4)(x + 9) (c) (x + 1)(x + 36) (d) (x + 1)(x + 12)
10. 10.Which is the correct factorisation of x² + 7x + 12? (a) (x + 3)(x + 4) (b) (x − 3)(x − 4) (c) (x + 1)(x + 12) (d) (x + 1)(x + 6)
11. 11.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8836 (b) 7396 (c) 8846 (d) 8116
12. 12.If a + b = 15 and ab = 56, what is (a − b)²? (a) 225 (b) 113 (c) 1 (d) 169
13. 13.If a + b = 6 and ab = 8, what is (a − b)²? (a) 36 (b) 2 (c) 20 (d) 4
14. 14.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 404 (b) 484 (c) 324 (d) 494
15. 15.If a + b = 15 and ab = 50, what is a² + b²? (a) 125 (b) 325 (c) 225 (d) 175
16. 16.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3136 (b) 4106 (c) 4096 (d) 3616
17. 17.What is the sum of the roots of x² − 9x + 20 = 0? (a) 9 (b) 8 (c) 10 (d) 20
18. 18.What are the roots of x² − 15x + 54 = 0? (a) 1 and 54 (b) 2 and 27 (c) 6 and 9 (d) 1 and 14
19. 19.What is the sum of the roots of x² − 13x + 42 = 0? (a) 14 (b) 12 (c) 13 (d) 42
20. 20.If a + b = 5 and ab = 6, what is (a − b)²? (a) 1 (b) 13 (c) 19 (d) 25
21. 21.If a + b = 11 and ab = 24, what is (a − b)²? (a) 25 (b) 121 (c) 73 (d) 5
22. 22.What is the sum of the roots of x² − 11x + 18 = 0? (a) 10 (b) 11 (c) 18 (d) 12
23. 23.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x + 6)(x + 9) (c) (x − 6)(x − 9) (d) (x + 1)(x + 54)
24. 24.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 657 (c) 441 (d) 225
25. 25.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 7921 (b) 8099 (c) 7931 (d) 8281
26. 26.If a + b = 8 and ab = 15, what is a² + b²? (a) 49 (b) 94 (c) 34 (d) 64
27. 27.If a + b = 7 and ab = 12, what is (a − b)²? (a) 37 (b) 25 (c) 1 (d) 49
28. 28.What is the sum of the roots of x² − 10x + 16 = 0? (a) 11 (b) 16 (c) 10 (d) 9
29. 29.If a + b = 18 and ab = 80, what is (a − b)²? (a) 2 (b) 164 (c) 4 (d) 324
30. 30.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1454 (b) 1444 (c) 1764 (d) 1596
31. 31.If a + b = 9 and ab = 18, what is a² + b²? (a) 117 (b) 81 (c) 45 (d) 63
32. 32.If a + b = 18 and ab = 72, what is a² + b²? (a) 468 (b) 252 (c) 324 (d) 180
33. 33.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 9604 (b) 10004 (c) 10414 (d) 10404
34. 34.If a + b = 11 and ab = 28, what is a² + b²? (a) 121 (b) 65 (c) 93 (d) 177
35. 35.What are the roots of x² − 16x + 63 = 0? (a) 1 and 15 (b) 1 and 63 (c) 3 and 21 (d) 7 and 9
36. 36.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 17 (d) 16
37. 37.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4235 (b) 4875 (c) 4225 (d) 5625
38. 38.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x − 4)(x − 5) (c) (x + 1)(x + 20) (d) (x + 1)(x + 8)
39. 39.What are the roots of x² − 13x + 40 = 0? (a) 1 and 40 (b) 1 and 12 (c) 2 and 20 (d) 5 and 8
40. 40.What is the sum of the roots of x² − 7x + 10 = 0? (a) 6 (b) 8 (c) 10 (d) 7

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