Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 61 · 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 89²? (a) 7931 (b) 8099 (c) 8281 (d) 7921
2. 2.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 16 (c) 72 (d) 17
3. 3.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x − 2)(x − 9) (c) (x + 1)(x + 18) (d) (x + 2)(x + 9)
4. 4.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2401 (b) 2411 (c) 2499 (d) 2601
5. 5.What is the product of the roots of x² − 10x + 21 = 0? (a) 10 (b) 22 (c) 20 (d) 21
6. 6.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4899 (b) 4771 (c) 4761 (d) 5041
7. 7.If a + b = 6 and ab = 8, what is a² + b²? (a) 28 (b) 36 (c) 20 (d) 52
8. 8.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8116 (b) 8846 (c) 8836 (d) 7396
9. 9.What is the product of the roots of x² − 8x + 12 = 0? (a) 13 (b) 11 (c) 8 (d) 12
10. 10.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x − 5)(x − 9) (c) (x + 5)(x + 9) (d) (x + 1)(x + 13)
11. 11.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1034 (b) 904 (c) 784 (d) 1024
12. 12.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 5)(x + 6) (c) (x + 1)(x + 10) (d) (x + 1)(x + 30)
13. 13.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 441 (c) 225 (d) 657
14. 14.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 9801 (b) 10211 (c) 10201 (d) 10001
15. 15.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x − 6)(x − 9) (c) (x + 1)(x + 54) (d) (x + 1)(x + 14)
16. 16.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 3035 (b) 2525 (c) 3025 (d) 2025
17. 17.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 16)
18. 18.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 739 (b) 729 (c) 1089 (d) 891
19. 19.If a + b = 12 and ab = 32, what is a² + b²? (a) 208 (b) 112 (c) 144 (d) 80
20. 20.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x + 5)(x + 7) (c) (x + 1)(x + 35) (d) (x − 5)(x − 7)
21. 21.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1764 (b) 1604 (c) 1444 (d) 1774
22. 22.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x − 7)(x − 9) (c) (x + 7)(x + 9) (d) (x + 1)(x + 15)
23. 23.What is the product of the roots of x² − 9x + 14 = 0? (a) 14 (b) 15 (c) 13 (d) 9
24. 24.If a + b = 9 and ab = 18, what is (a − b)²? (a) 3 (b) 9 (c) 81 (d) 45
25. 25.What is the sum of the roots of x² − 7x + 12 = 0? (a) 12 (b) 6 (c) 7 (d) 8
26. 26.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1531 (b) 1521 (c) 1599 (d) 1681
27. 27.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3146 (b) 3136 (c) 4096 (d) 3584
28. 28.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8109 (b) 8649 (c) 7569 (d) 8659
29. 29.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
30. 30.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
31. 31.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1156 (b) 676 (c) 1166 (d) 916
32. 32.If a + b = 7 and ab = 10, what is (a − b)²? (a) 3 (b) 9 (c) 29 (d) 49
33. 33.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 396 (b) 324 (c) 484 (d) 334
34. 34.If a + b = 3 and ab = 2, what is a² + b²? (a) 13 (b) 9 (c) 5 (d) 7
35. 35.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 266 (b) 256 (c) 384 (d) 576
36. 36.What are the roots of x² − 14x + 48 = 0? (a) 1 and 48 (b) 6 and 8 (c) 1 and 13 (d) 2 and 24
37. 37.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2704 (b) 2496 (c) 2314 (d) 2304
38. 38.If a + b = 14 and ab = 48, what is a² + b²? (a) 196 (b) 292 (c) 100 (d) 148
39. 39.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7056 (b) 5776 (c) 6416 (d) 7066
40. 40.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6724 (b) 6734 (c) 6404 (d) 6084

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