Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 674 · 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.If a + b = 6 and ab = 5, what is a² + b²? (a) 36 (b) 46 (c) 26 (d) 31
2. 2.If a + b = 11 and ab = 30, what is (a − b)²? (a) 1 (b) 121 (c) 91 (d) 61
3. 3.If a + b = 13 and ab = 30, what is (a − b)²? (a) 109 (b) 7 (c) 169 (d) 49
4. 4.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6571 (b) 6401 (c) 6241 (d) 6561
5. 5.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)
6. 6.If a + b = 14 and ab = 24, what is (a − b)²? (a) 100 (b) 10 (c) 196 (d) 148
7. 7.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1591 (b) 1849 (c) 1379 (d) 1369
8. 8.If a + b = 18 and ab = 80, what is a² + b²? (a) 164 (b) 484 (c) 324 (d) 244
9. 9.If a + b = 13 and ab = 42, what is a² + b²? (a) 127 (b) 253 (c) 169 (d) 85
10. 10.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 1 and 14 (c) 2 and 28 (d) 7 and 8
11. 11.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 625 (b) 635 (c) 225 (d) 425
12. 12.What are the roots of x² − 11x + 30 = 0? (a) 5 and 6 (b) 1 and 30 (c) 2 and 15 (d) 1 and 10
13. 13.What are the roots of x² − 14x + 48 = 0? (a) 1 and 13 (b) 6 and 8 (c) 2 and 24 (d) 1 and 48
14. 14.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 971 (b) 901 (c) 961 (d) 841
15. 15.If a + b = 21 and ab = 108, what is (a − b)²? (a) 3 (b) 441 (c) 9 (d) 225
16. 16.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 3)(x + 5) (c) (x + 1)(x + 7) (d) (x + 1)(x + 15)
17. 17.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 1)(x + 35) (c) (x + 1)(x + 11) (d) (x + 5)(x + 7)
18. 18.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x − 7)(x − 8) (c) (x + 1)(x + 56) (d) (x + 7)(x + 8)
19. 19.What are the roots of x² − 12x + 32 = 0? (a) 2 and 16 (b) 1 and 11 (c) 1 and 32 (d) 4 and 8
20. 20.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 1 and 16 (c) 1 and 72 (d) 8 and 9
21. 21.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 1)(x + 15) (c) (x + 7)(x + 9) (d) (x − 7)(x − 9)
22. 22.If a + b = 18 and ab = 72, what is a² + b²? (a) 324 (b) 468 (c) 180 (d) 252
23. 23.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 12) (c) (x + 1)(x + 40) (d) (x + 5)(x + 8)
24. 24.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 3035 (b) 2025 (c) 2525 (d) 3025
25. 25.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 9409 (b) 10009 (c) 10619 (d) 10609
26. 26.If a + b = 14 and ab = 33, what is (a − b)²? (a) 8 (b) 130 (c) 64 (d) 196
27. 27.If a + b = 18 and ab = 77, what is a² + b²? (a) 247 (b) 324 (c) 478 (d) 170
28. 28.If a + b = 16 and ab = 55, what is a² + b²? (a) 366 (b) 201 (c) 256 (d) 146
29. 29.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 5476 (b) 4884 (c) 4366 (d) 4356
30. 30.What is the product of the roots of x² − 13x + 36 = 0? (a) 37 (b) 35 (c) 13 (d) 36
31. 31.What are the roots of x² − 12x + 27 = 0? (a) 3 and 9 (b) 1 and 27 (c) 1 and 11 (d) 2 and 10
32. 32.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 4225 (b) 3025 (c) 3575 (d) 3035
33. 33.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2401 (b) 2501 (c) 2611 (d) 2601
34. 34.What is the sum of the roots of x² − 11x + 18 = 0? (a) 12 (b) 18 (c) 10 (d) 11
35. 35.If a + b = 14 and ab = 24, what is a² + b²? (a) 148 (b) 196 (c) 244 (d) 172
36. 36.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 8836 (b) 7396 (c) 7406 (d) 8084
37. 37.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 484 (b) 494 (c) 404 (d) 324
38. 38.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 371 (b) 361 (c) 441 (d) 399
39. 39.What is the product of the roots of x² − 12x + 27 = 0? (a) 28 (b) 26 (c) 27 (d) 12
40. 40.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1599 (b) 1521 (c) 1681 (d) 1531

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