Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 202 · 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.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3616 (b) 3136 (c) 4096 (d) 4106
2. 2.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 409 (b) 289 (c) 529 (d) 539
3. 3.What is the product of the roots of x² − 15x + 54 = 0? (a) 15 (b) 53 (c) 54 (d) 55
4. 4.What are the roots of x² − 8x + 12 = 0? (a) 1 and 12 (b) 2 and 6 (c) 3 and 4 (d) 1 and 7
5. 5.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3604 (b) 3844 (c) 3854 (d) 3364
6. 6.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 56 (c) 15 (d) 14
7. 7.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 73 (c) 72 (d) 71
8. 8.If a + b = 13 and ab = 22, what is a² + b²? (a) 213 (b) 147 (c) 125 (d) 169
9. 9.What are the roots of x² − 10x + 24 = 0? (a) 1 and 24 (b) 4 and 6 (c) 2 and 12 (d) 1 and 9
10. 10.If a + b = 8 and ab = 12, what is a² + b²? (a) 40 (b) 88 (c) 52 (d) 64
11. 11.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 5194 (b) 4624 (c) 5184 (d) 4904
12. 12.What is the sum of the roots of x² − 7x + 10 = 0? (a) 6 (b) 10 (c) 8 (d) 7
13. 13.If a + b = 14 and ab = 48, what is a² + b²? (a) 100 (b) 292 (c) 196 (d) 148
14. 14.If a + b = 14 and ab = 33, what is (a − b)²? (a) 196 (b) 8 (c) 64 (d) 130
15. 15.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x + 1)(x + 10) (c) (x − 4)(x − 7) (d) (x + 4)(x + 7)
16. 16.If a + b = 20 and ab = 99, what is (a − b)²? (a) 4 (b) 202 (c) 400 (d) 2
17. 17.If a + b = 12 and ab = 32, what is a² + b²? (a) 80 (b) 144 (c) 208 (d) 112
18. 18.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5041 (b) 5051 (c) 4901 (d) 4761
19. 19.If a + b = 14 and ab = 45, what is (a − b)²? (a) 4 (b) 106 (c) 196 (d) 16
20. 20.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1601 (b) 1691 (c) 1681 (d) 1521
21. 21.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)
22. 22.Which is the correct factorisation of x² + 9x + 14? (a) (x + 2)(x + 7) (b) (x + 1)(x + 8) (c) (x − 2)(x − 7) (d) (x + 1)(x + 14)
23. 23.If a + b = 17 and ab = 72, what is (a − b)²? (a) 145 (b) 289 (c) 217 (d) 1
24. 24.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9811 (b) 9999 (c) 9801 (d) 10201
25. 25.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4896 (b) 5184 (c) 4624 (d) 4634
26. 26.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6404 (b) 6084 (c) 6724 (d) 6734
27. 27.Which is the correct factorisation of x² + 5x + 6? (a) (x + 2)(x + 3) (b) (x − 2)(x − 3) (c) (x + 1)(x + 4) (d) (x + 1)(x + 6)
28. 28.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 9 (c) 3 (d) 225
29. 29.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
30. 30.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1584 (b) 1296 (c) 1306 (d) 1936
31. 31.Which is the correct factorisation of x² + 9x + 18? (a) (x + 3)(x + 6) (b) (x + 1)(x + 18) (c) (x + 1)(x + 8) (d) (x − 3)(x − 6)
32. 32.If a + b = 16 and ab = 60, what is (a − b)²? (a) 136 (b) 256 (c) 4 (d) 16
33. 33.Which is the correct factorisation of x² + 7x + 12? (a) (x + 3)(x + 4) (b) (x + 1)(x + 12) (c) (x + 1)(x + 6) (d) (x − 3)(x − 4)
34. 34.If a + b = 8 and ab = 7, what is a² + b²? (a) 50 (b) 64 (c) 78 (d) 57
35. 35.What is the product of the roots of x² − 10x + 16 = 0? (a) 15 (b) 16 (c) 17 (d) 10
36. 36.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x + 1)(x + 10) (c) (x + 3)(x + 8) (d) (x − 3)(x − 8)
37. 37.What is the sum of the roots of x² − 5x + 6 = 0? (a) 6 (b) 7 (c) 5 (d) 4
38. 38.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 5929 (b) 6391 (c) 6889 (d) 5939
39. 39.What are the roots of x² − 9x + 18 = 0? (a) 1 and 18 (b) 2 and 9 (c) 3 and 6 (d) 1 and 8
40. 40.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 441 (b) 371 (c) 361 (d) 399

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