Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 539 · 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 = 19 and ab = 90, what is a² + b²? (a) 541 (b) 271 (c) 361 (d) 181
2. 2.What is the sum of the roots of x² − 10x + 24 = 0? (a) 10 (b) 24 (c) 11 (d) 9
3. 3.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 1)(x + 12) (c) (x + 4)(x + 9) (d) (x + 1)(x + 36)
4. 4.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 63 (c) 17 (d) 15
5. 5.What are the roots of x² − 13x + 42 = 0? (a) 1 and 42 (b) 1 and 12 (c) 6 and 7 (d) 2 and 21
6. 6.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 4096 (b) 3136 (c) 3616 (d) 4106
7. 7.What is the sum of the roots of x² − 14x + 48 = 0? (a) 15 (b) 14 (c) 13 (d) 48
8. 8.If a + b = 20 and ab = 96, what is (a − b)²? (a) 4 (b) 400 (c) 16 (d) 208
9. 9.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 1 and 16 (c) 8 and 9 (d) 2 and 36
10. 10.Which is the correct factorisation of x² + 8x + 12? (a) (x − 2)(x − 6) (b) (x + 1)(x + 7) (c) (x + 1)(x + 12) (d) (x + 2)(x + 6)
11. 11.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9216 (b) 10816 (c) 9226 (d) 9984
12. 12.What is the sum of the roots of x² − 13x + 40 = 0? (a) 40 (b) 14 (c) 12 (d) 13
13. 13.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 4)(x + 7) (c) (x + 1)(x + 10) (d) (x + 1)(x + 28)
14. 14.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1936 (b) 1306 (c) 1584 (d) 1296
15. 15.If a + b = 11 and ab = 18, what is a² + b²? (a) 157 (b) 121 (c) 85 (d) 103
16. 16.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 361 (b) 441 (c) 371 (d) 399
17. 17.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2704 (b) 2304 (c) 2504 (d) 2714
18. 18.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 896 (c) 1024 (d) 784
19. 19.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 9025 (b) 11035 (c) 10025 (d) 11025
20. 20.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)
21. 21.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 8)(x + 9) (c) (x + 1)(x + 16) (d) (x − 8)(x − 9)
22. 22.What are the roots of x² − 11x + 18 = 0? (a) 2 and 9 (b) 1 and 10 (c) 3 and 6 (d) 1 and 18
23. 23.Which is the correct factorisation of x² + 8x + 15? (a) (x + 3)(x + 5) (b) (x + 1)(x + 15) (c) (x − 3)(x − 5) (d) (x + 1)(x + 7)
24. 24.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4489 (b) 4499 (c) 4891 (d) 5329
25. 25.What is the sum of the roots of x² − 11x + 28 = 0? (a) 28 (b) 11 (c) 10 (d) 12
26. 26.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2035 (b) 2025 (c) 1225 (d) 1625
27. 27.What are the roots of x² − 10x + 21 = 0? (a) 1 and 9 (b) 2 and 8 (c) 1 and 21 (d) 3 and 7
28. 28.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 916 (b) 676 (c) 1156 (d) 1166
29. 29.If a + b = 15 and ab = 54, what is (a − b)²? (a) 225 (b) 3 (c) 117 (d) 9
30. 30.If a + b = 10 and ab = 16, what is (a − b)²? (a) 68 (b) 36 (c) 100 (d) 6
31. 31.If a + b = 7 and ab = 12, what is a² + b²? (a) 73 (b) 37 (c) 25 (d) 49
32. 32.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x + 1)(x + 16) (c) (x + 2)(x + 8) (d) (x − 2)(x − 8)
33. 33.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 925 (b) 625 (c) 1225 (d) 1235
34. 34.If a + b = 19 and ab = 88, what is a² + b²? (a) 537 (b) 361 (c) 273 (d) 185
35. 35.If a + b = 13 and ab = 42, what is a² + b²? (a) 127 (b) 253 (c) 85 (d) 169
36. 36.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 3025 (b) 2035 (c) 2475 (d) 2025
37. 37.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 9801 (b) 10201 (c) 10211 (d) 10001
38. 38.If a + b = 9 and ab = 18, what is a² + b²? (a) 117 (b) 81 (c) 63 (d) 45
39. 39.If a + b = 7 and ab = 6, what is a² + b²? (a) 61 (b) 37 (c) 43 (d) 49
40. 40.If a + b = 12 and ab = 11, what is (a − b)²? (a) 144 (b) 122 (c) 100 (d) 10

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