Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 862 · 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) 4096 (b) 3616 (c) 4106 (d) 3136
2. 2.What is the sum of the roots of x² − 15x + 54 = 0? (a) 16 (b) 15 (c) 14 (d) 54
3. 3.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 334 (b) 324 (c) 396 (d) 484
4. 4.If a + b = 17 and ab = 70, what is a² + b²? (a) 219 (b) 149 (c) 429 (d) 289
5. 5.If a + b = 15 and ab = 56, what is (a − b)²? (a) 1 (b) 225 (c) 113 (d) 169
6. 6.What is the product of the roots of x² − 11x + 28 = 0? (a) 29 (b) 11 (c) 27 (d) 28
7. 7.If a + b = 20 and ab = 96, what is a² + b²? (a) 400 (b) 592 (c) 208 (d) 304
8. 8.If a + b = 14 and ab = 48, what is a² + b²? (a) 196 (b) 148 (c) 100 (d) 292
9. 9.If a + b = 17 and ab = 70, what is (a − b)²? (a) 289 (b) 3 (c) 9 (d) 149
10. 10.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 7569 (b) 8109 (c) 8659 (d) 8649
11. 11.What is the sum of the roots of x² − 7x + 12 = 0? (a) 8 (b) 6 (c) 12 (d) 7
12. 12.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x + 3)(x + 7) (c) (x − 3)(x − 7) (d) (x + 1)(x + 9)
13. 13.What are the roots of x² − 9x + 20 = 0? (a) 1 and 8 (b) 2 and 10 (c) 4 and 5 (d) 1 and 20
14. 14.If a + b = 16 and ab = 48, what is a² + b²? (a) 160 (b) 256 (c) 352 (d) 208
15. 15.If a + b = 12 and ab = 20, what is a² + b²? (a) 144 (b) 184 (c) 104 (d) 124
16. 16.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1099 (b) 909 (c) 1089 (d) 729
17. 17.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10211 (b) 10201 (c) 9801 (d) 10001
18. 18.What are the roots of x² − 8x + 15 = 0? (a) 2 and 6 (b) 3 and 5 (c) 1 and 15 (d) 1 and 7
19. 19.What are the roots of x² − 10x + 21 = 0? (a) 1 and 9 (b) 3 and 7 (c) 1 and 21 (d) 2 and 8
20. 20.What is the product of the roots of x² − 6x + 8 = 0? (a) 6 (b) 8 (c) 9 (d) 7
21. 21.If a + b = 15 and ab = 56, what is a² + b²? (a) 337 (b) 113 (c) 225 (d) 169
22. 22.If a + b = 17 and ab = 66, what is a² + b²? (a) 421 (b) 157 (c) 289 (d) 223
23. 23.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 971 (b) 841 (c) 901 (d) 961
24. 24.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1225 (b) 2025 (c) 1235 (d) 1575
25. 25.If a + b = 19 and ab = 88, what is a² + b²? (a) 185 (b) 361 (c) 273 (d) 537
26. 26.What is the sum of the roots of x² − 9x + 20 = 0? (a) 20 (b) 10 (c) 9 (d) 8
27. 27.If a + b = 13 and ab = 40, what is a² + b²? (a) 169 (b) 129 (c) 249 (d) 89
28. 28.If a + b = 16 and ab = 60, what is (a − b)²? (a) 4 (b) 16 (c) 256 (d) 136
29. 29.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
30. 30.What is the sum of the roots of x² − 11x + 30 = 0? (a) 11 (b) 12 (c) 10 (d) 30
31. 31.If a + b = 10 and ab = 24, what is a² + b²? (a) 100 (b) 148 (c) 76 (d) 52
32. 32.If a + b = 14 and ab = 33, what is a² + b²? (a) 196 (b) 163 (c) 130 (d) 262
33. 33.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1521 (b) 1691 (c) 1601 (d) 1681
34. 34.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1584 (b) 1936 (c) 1306 (d) 1296
35. 35.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 961 (b) 841 (c) 851 (d) 899
36. 36.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9999 (b) 10201 (c) 9811 (d) 9801
37. 37.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 17 (c) 16 (d) 15
38. 38.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2809 (b) 2219 (c) 2209 (d) 2491
39. 39.If a + b = 11 and ab = 18, what is (a − b)²? (a) 121 (b) 85 (c) 49 (d) 7
40. 40.If a + b = 19 and ab = 90, what is (a − b)²? (a) 1 (b) 181 (c) 361 (d) 271

Free maths olympiad lessons, practice and worksheets at talentjr.in/maths-olympiad