Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 416 · 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.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 11) (b) (x − 4)(x − 8) (c) (x + 1)(x + 32) (d) (x + 4)(x + 8)
2. 2.What are the roots of x² − 10x + 21 = 0? (a) 1 and 21 (b) 1 and 9 (c) 2 and 8 (d) 3 and 7
3. 3.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x − 4)(x − 7) (c) (x + 4)(x + 7) (d) (x + 1)(x + 10)
4. 4.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 1)(x + 9) (c) (x + 4)(x + 6) (d) (x + 1)(x + 24)
5. 5.If a + b = 18 and ab = 72, what is (a − b)²? (a) 180 (b) 36 (c) 324 (d) 6
6. 6.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 271 (c) 181 (d) 541
7. 7.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1604 (b) 1764 (c) 1444 (d) 1774
8. 8.If a + b = 17 and ab = 60, what is (a − b)²? (a) 49 (b) 169 (c) 7 (d) 289
9. 9.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4904 (b) 4624 (c) 5184 (d) 5194
10. 10.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1609 (b) 1849 (c) 1859 (d) 1369
11. 11.If a + b = 17 and ab = 72, what is (a − b)²? (a) 1 (b) 145 (c) 217 (d) 289
12. 12.If a + b = 19 and ab = 84, what is (a − b)²? (a) 361 (b) 5 (c) 193 (d) 25
13. 13.If a + b = 18 and ab = 77, what is a² + b²? (a) 324 (b) 170 (c) 478 (d) 247
14. 14.If a + b = 9 and ab = 18, what is (a − b)²? (a) 9 (b) 45 (c) 3 (d) 81
15. 15.Which is the correct factorisation of x² + 11x + 24? (a) (x + 3)(x + 8) (b) (x + 1)(x + 24) (c) (x + 1)(x + 10) (d) (x − 3)(x − 8)
16. 16.If a + b = 11 and ab = 28, what is (a − b)²? (a) 9 (b) 65 (c) 121 (d) 3
17. 17.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 384 (b) 266 (c) 256 (d) 576
18. 18.If a + b = 11 and ab = 30, what is a² + b²? (a) 181 (b) 91 (c) 61 (d) 121
19. 19.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9975 (b) 9025 (c) 11025 (d) 9035
20. 20.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 7225 (b) 8125 (c) 9025 (d) 9035
21. 21.If a + b = 21 and ab = 108, what is a² + b²? (a) 225 (b) 333 (c) 441 (d) 657
22. 22.If a + b = 16 and ab = 55, what is (a − b)²? (a) 36 (b) 256 (c) 6 (d) 146
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.What is the product of the roots of x² − 9x + 20 = 0? (a) 9 (b) 21 (c) 20 (d) 19
25. 25.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10619 (b) 10609 (c) 10009 (d) 9409
26. 26.If a + b = 6 and ab = 5, what is a² + b²? (a) 31 (b) 46 (c) 36 (d) 26
27. 27.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 1)(x + 30) (c) (x + 1)(x + 10) (d) (x + 5)(x + 6)
28. 28.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1306 (b) 1584 (c) 1936 (d) 1296
29. 29.If a + b = 20 and ab = 96, what is (a − b)²? (a) 400 (b) 208 (c) 16 (d) 4
30. 30.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 4225 (b) 3575 (c) 3035 (d) 3025
31. 31.If a + b = 9 and ab = 14, what is a² + b²? (a) 109 (b) 81 (c) 53 (d) 67
32. 32.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10004 (b) 10404 (c) 10414 (d) 9604
33. 33.What is the sum of the roots of x² − 10x + 24 = 0? (a) 10 (b) 24 (c) 9 (d) 11
34. 34.What are the roots of x² − 11x + 30 = 0? (a) 1 and 30 (b) 2 and 15 (c) 5 and 6 (d) 1 and 10
35. 35.If a + b = 13 and ab = 30, what is (a − b)²? (a) 169 (b) 49 (c) 109 (d) 7
36. 36.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
37. 37.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6899 (b) 6409 (c) 6889 (d) 5929
38. 38.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 4 (c) 202 (d) 2
39. 39.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
40. 40.What are the roots of x² − 7x + 12 = 0? (a) 3 and 4 (b) 2 and 6 (c) 1 and 6 (d) 1 and 12

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