Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 876 · 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 = 9 and ab = 20, what is a² + b²? (a) 41 (b) 121 (c) 81 (d) 61
2. 2.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x + 1)(x + 54) (c) (x + 1)(x + 14) (d) (x − 6)(x − 9)
3. 3.If a + b = 16 and ab = 55, what is a² + b²? (a) 256 (b) 366 (c) 201 (d) 146
4. 4.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 361 (b) 401 (c) 441 (d) 451
5. 5.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x − 2)(x − 6) (c) (x + 2)(x + 6) (d) (x + 1)(x + 12)
6. 6.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4891 (b) 5329 (c) 4489 (d) 4499
7. 7.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)
8. 8.What is the sum of the roots of x² − 11x + 24 = 0? (a) 10 (b) 24 (c) 12 (d) 11
9. 9.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 15 (c) 16 (d) 63
10. 10.If a + b = 11 and ab = 28, what is a² + b²? (a) 65 (b) 93 (c) 177 (d) 121
11. 11.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2126 (b) 2916 (c) 2116 (d) 2484
12. 12.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 541 (c) 181 (d) 361
13. 13.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x − 4)(x − 5) (c) (x + 1)(x + 20) (d) (x + 4)(x + 5)
14. 14.Which is the correct factorisation of x² + 11x + 18? (a) (x + 2)(x + 9) (b) (x + 1)(x + 18) (c) (x − 2)(x − 9) (d) (x + 1)(x + 10)
15. 15.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 1)(x + 13) (c) (x + 5)(x + 9) (d) (x + 1)(x + 45)
16. 16.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 1)(x + 30) (c) (x + 5)(x + 6) (d) (x − 5)(x − 6)
17. 17.Which is the correct factorisation of x² + 12x + 35? (a) (x + 5)(x + 7) (b) (x + 1)(x + 11) (c) (x + 1)(x + 35) (d) (x − 5)(x − 7)
18. 18.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 56 (c) 55 (d) 57
19. 19.If a + b = 13 and ab = 12, what is (a − b)²? (a) 145 (b) 121 (c) 169 (d) 11
20. 20.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x − 7)(x − 8) (d) (x + 7)(x + 8)
21. 21.What is the product of the roots of x² − 14x + 48 = 0? (a) 47 (b) 14 (c) 49 (d) 48
22. 22.If a + b = 11 and ab = 18, what is a² + b²? (a) 121 (b) 157 (c) 85 (d) 103
23. 23.Which is the correct factorisation of x² + 5x + 6? (a) (x + 2)(x + 3) (b) (x + 1)(x + 6) (c) (x − 2)(x − 3) (d) (x + 1)(x + 4)
24. 24.If a + b = 16 and ab = 55, what is (a − b)²? (a) 36 (b) 256 (c) 146 (d) 6
25. 25.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 7921 (b) 7931 (c) 8281 (d) 8099
26. 26.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6734 (b) 6404 (c) 6724 (d) 6084
27. 27.If a + b = 21 and ab = 108, what is (a − b)²? (a) 3 (b) 9 (c) 225 (d) 441
28. 28.If a + b = 20 and ab = 96, what is a² + b²? (a) 208 (b) 400 (c) 592 (d) 304
29. 29.Which is the correct factorisation of x² + 10x + 16? (a) (x + 2)(x + 8) (b) (x + 1)(x + 9) (c) (x − 2)(x − 8) (d) (x + 1)(x + 16)
30. 30.If a + b = 11 and ab = 10, what is a² + b²? (a) 121 (b) 101 (c) 141 (d) 111
31. 31.What is the product of the roots of x² − 11x + 30 = 0? (a) 11 (b) 31 (c) 30 (d) 29
32. 32.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 1 and 72 (c) 2 and 36 (d) 8 and 9
33. 33.If a + b = 11 and ab = 30, what is a² + b²? (a) 91 (b) 61 (c) 121 (d) 181
34. 34.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 409 (b) 539 (c) 529 (d) 289
35. 35.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 7569 (b) 7579 (c) 8091 (d) 8649
36. 36.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1936 (b) 1306 (c) 1584 (d) 1296
37. 37.What is the product of the roots of x² − 11x + 24 = 0? (a) 11 (b) 23 (c) 24 (d) 25
38. 38.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 17 (c) 16 (d) 72
39. 39.What is the product of the roots of x² − 9x + 14 = 0? (a) 15 (b) 9 (c) 13 (d) 14
40. 40.Which is the correct factorisation of x² + 10x + 24? (a) (x + 4)(x + 6) (b) (x + 1)(x + 24) (c) (x − 4)(x − 6) (d) (x + 1)(x + 9)

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