Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 444 · 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 sum of the roots of x² − 15x + 54 = 0? (a) 54 (b) 15 (c) 16 (d) 14
2. 2.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6391 (b) 5929 (c) 5939 (d) 6889
3. 3.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x + 4)(x + 5) (c) (x − 4)(x − 5) (d) (x + 1)(x + 8)
4. 4.What is the sum of the roots of x² − 11x + 28 = 0? (a) 12 (b) 28 (c) 11 (d) 10
5. 5.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 404 (b) 484 (c) 494 (d) 324
6. 6.If a + b = 16 and ab = 60, what is a² + b²? (a) 256 (b) 196 (c) 136 (d) 376
7. 7.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 72 (c) 17 (d) 71
8. 8.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 1 and 72 (c) 8 and 9 (d) 1 and 16
9. 9.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 3)(x + 4) (c) (x + 1)(x + 6) (d) (x + 1)(x + 12)
10. 10.Which is the correct factorisation of x² + 9x + 18? (a) (x + 3)(x + 6) (b) (x − 3)(x − 6) (c) (x + 1)(x + 18) (d) (x + 1)(x + 8)
11. 11.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 72 (c) 16 (d) 17
12. 12.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6734 (b) 6404 (c) 6084 (d) 6724
13. 13.Which is the correct factorisation of x² + 12x + 35? (a) (x + 5)(x + 7) (b) (x − 5)(x − 7) (c) (x + 1)(x + 35) (d) (x + 1)(x + 11)
14. 14.If a + b = 15 and ab = 44, what is (a − b)²? (a) 49 (b) 7 (c) 225 (d) 137
15. 15.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 18) (b) (x + 2)(x + 9) (c) (x − 2)(x − 9) (d) (x + 1)(x + 10)
16. 16.Which is the correct factorisation of x² + 11x + 24? (a) (x + 3)(x + 8) (b) (x + 1)(x + 10) (c) (x + 1)(x + 24) (d) (x − 3)(x − 8)
17. 17.What is the product of the roots of x² − 12x + 35 = 0? (a) 36 (b) 12 (c) 35 (d) 34
18. 18.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 10609 (b) 9409 (c) 9991 (d) 9419
19. 19.If a + b = 10 and ab = 16, what is (a − b)²? (a) 6 (b) 100 (c) 68 (d) 36
20. 20.If a + b = 11 and ab = 30, what is (a − b)²? (a) 61 (b) 1 (c) 121 (d) 91
21. 21.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2035 (b) 3025 (c) 2475 (d) 2025
22. 22.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 401 (b) 361 (c) 451 (d) 441
23. 23.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1369 (b) 1609 (c) 1859 (d) 1849
24. 24.If a + b = 11 and ab = 28, what is (a − b)²? (a) 3 (b) 9 (c) 121 (d) 65
25. 25.If a + b = 18 and ab = 80, what is (a − b)²? (a) 4 (b) 324 (c) 2 (d) 164
26. 26.What is the product of the roots of x² − 11x + 30 = 0? (a) 31 (b) 29 (c) 11 (d) 30
27. 27.If a + b = 6 and ab = 5, what is a² + b²? (a) 46 (b) 26 (c) 36 (d) 31
28. 28.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x + 1)(x + 36) (c) (x − 4)(x − 9) (d) (x + 4)(x + 9)
29. 29.If a + b = 12 and ab = 27, what is (a − b)²? (a) 36 (b) 90 (c) 144 (d) 6
30. 30.What are the roots of x² − 11x + 24 = 0? (a) 2 and 12 (b) 1 and 10 (c) 1 and 24 (d) 3 and 8
31. 31.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 9 (c) 441 (d) 3
32. 32.If a + b = 19 and ab = 88, what is (a − b)²? (a) 3 (b) 185 (c) 361 (d) 9
33. 33.If a + b = 16 and ab = 63, what is (a − b)²? (a) 256 (b) 4 (c) 2 (d) 130
34. 34.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 8) (b) (x + 1)(x + 14) (c) (x + 2)(x + 7) (d) (x − 2)(x − 7)
35. 35.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 7225 (b) 5625 (c) 6375 (d) 5635
36. 36.If a + b = 15 and ab = 54, what is (a − b)²? (a) 225 (b) 9 (c) 3 (d) 117
37. 37.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 409 (b) 289 (c) 529 (d) 539
38. 38.If a + b = 17 and ab = 60, what is a² + b²? (a) 169 (b) 409 (c) 229 (d) 289
39. 39.What are the roots of x² − 10x + 16 = 0? (a) 2 and 8 (b) 1 and 9 (c) 3 and 7 (d) 1 and 16
40. 40.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 56) (c) (x + 1)(x + 14) (d) (x + 7)(x + 8)

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