Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 454 · 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.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 17 (c) 72 (d) 71
2. 2.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1936 (b) 1584 (c) 1306 (d) 1296
3. 3.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9991 (b) 9419 (c) 10609 (d) 9409
4. 4.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 5625 (b) 7235 (c) 7225 (d) 6425
5. 5.What is the sum of the roots of x² − 11x + 28 = 0? (a) 28 (b) 11 (c) 10 (d) 12
6. 6.If a + b = 13 and ab = 40, what is a² + b²? (a) 89 (b) 129 (c) 249 (d) 169
7. 7.If a + b = 13 and ab = 30, what is (a − b)²? (a) 109 (b) 7 (c) 49 (d) 169
8. 8.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 676 (b) 916 (c) 1156 (d) 1166
9. 9.What are the roots of x² − 10x + 16 = 0? (a) 3 and 7 (b) 1 and 9 (c) 2 and 8 (d) 1 and 16
10. 10.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 1 and 72 (c) 2 and 36 (d) 1 and 16
11. 11.If a + b = 19 and ab = 90, what is (a − b)²? (a) 1 (b) 181 (c) 361 (d) 271
12. 12.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 8)(x + 9) (c) (x + 1)(x + 16) (d) (x + 1)(x + 72)
13. 13.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 1)(x + 10) (c) (x + 5)(x + 6) (d) (x + 1)(x + 30)
14. 14.What is the sum of the roots of x² − 14x + 48 = 0? (a) 14 (b) 15 (c) 13 (d) 48
15. 15.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 576 (b) 256 (c) 416 (d) 586
16. 16.If a + b = 15 and ab = 50, what is a² + b²? (a) 175 (b) 125 (c) 325 (d) 225
17. 17.If a + b = 10 and ab = 16, what is a² + b²? (a) 132 (b) 100 (c) 84 (d) 68
18. 18.What is the sum of the roots of x² − 13x + 36 = 0? (a) 13 (b) 12 (c) 14 (d) 36
19. 19.If a + b = 17 and ab = 70, what is a² + b²? (a) 219 (b) 289 (c) 429 (d) 149
20. 20.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3625 (b) 4235 (c) 3025 (d) 4225
21. 21.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 56) (c) (x + 7)(x + 8) (d) (x + 1)(x + 14)
22. 22.If a + b = 15 and ab = 56, what is a² + b²? (a) 337 (b) 225 (c) 169 (d) 113
23. 23.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 235 (b) 625 (c) 375 (d) 225
24. 24.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 6375 (b) 7225 (c) 5625 (d) 5635
25. 25.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2819 (b) 2809 (c) 2209 (d) 2509
26. 26.What is the product of the roots of x² − 10x + 21 = 0? (a) 21 (b) 20 (c) 22 (d) 10
27. 27.If a + b = 18 and ab = 77, what is a² + b²? (a) 324 (b) 170 (c) 478 (d) 247
28. 28.If a + b = 4 and ab = 3, what is a² + b²? (a) 16 (b) 13 (c) 10 (d) 22
29. 29.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x − 7)(x − 9) (c) (x + 7)(x + 9) (d) (x + 1)(x + 63)
30. 30.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 4)(x + 6) (c) (x + 1)(x + 9) (d) (x + 1)(x + 24)
31. 31.What is the sum of the roots of x² − 11x + 24 = 0? (a) 11 (b) 24 (c) 12 (d) 10
32. 32.What is the sum of the roots of x² − 13x + 42 = 0? (a) 14 (b) 13 (c) 42 (d) 12
33. 33.If a + b = 7 and ab = 10, what is a² + b²? (a) 29 (b) 49 (c) 39 (d) 69
34. 34.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6084 (b) 6094 (c) 6396 (d) 6724
35. 35.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x + 1)(x + 13) (c) (x + 5)(x + 9) (d) (x − 5)(x − 9)
36. 36.If a + b = 20 and ab = 99, what is (a − b)²? (a) 4 (b) 2 (c) 202 (d) 400
37. 37.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 441 (b) 399 (c) 361 (d) 371
38. 38.If a + b = 15 and ab = 56, what is (a − b)²? (a) 113 (b) 225 (c) 169 (d) 1
39. 39.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8649 (b) 8109 (c) 7569 (d) 8659
40. 40.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4761 (b) 5041 (c) 4899 (d) 4771

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