Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 922 · 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 product of the roots of x² − 13x + 42 = 0? (a) 13 (b) 43 (c) 41 (d) 42
2. 2.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x − 4)(x − 5) (c) (x + 4)(x + 5) (d) (x + 1)(x + 20)
3. 3.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x + 1)(x + 12) (c) (x + 5)(x + 8) (d) (x − 5)(x − 8)
4. 4.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8846 (b) 7396 (c) 8836 (d) 8116
5. 5.Which is the correct factorisation of x² + 12x + 27? (a) (x + 3)(x + 9) (b) (x + 1)(x + 11) (c) (x − 3)(x − 9) (d) (x + 1)(x + 27)
6. 6.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2499 (b) 2401 (c) 2411 (d) 2601
7. 7.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 11025 (b) 9975 (c) 9035 (d) 9025
8. 8.If a + b = 11 and ab = 18, what is a² + b²? (a) 121 (b) 103 (c) 157 (d) 85
9. 9.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 1 and 72 (c) 1 and 16 (d) 2 and 36
10. 10.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 6)(x + 7) (c) (x + 1)(x + 42) (d) (x + 1)(x + 12)
11. 11.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 71 (c) 72 (d) 17
12. 12.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4225 (b) 4235 (c) 5625 (d) 4875
13. 13.What is the sum of the roots of x² − 14x + 48 = 0? (a) 14 (b) 15 (c) 48 (d) 13
14. 14.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6899 (b) 6409 (c) 6889 (d) 5929
15. 15.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7235 (b) 6425 (c) 7225 (d) 5625
16. 16.What is the sum of the roots of x² − 15x + 54 = 0? (a) 54 (b) 16 (c) 15 (d) 14
17. 17.Which is the correct factorisation of x² + 10x + 16? (a) (x + 2)(x + 8) (b) (x + 1)(x + 16) (c) (x + 1)(x + 9) (d) (x − 2)(x − 8)
18. 18.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3854 (b) 3844 (c) 3364 (d) 3604
19. 19.What is the product of the roots of x² − 13x + 36 = 0? (a) 13 (b) 36 (c) 35 (d) 37
20. 20.If a + b = 14 and ab = 33, what is a² + b²? (a) 163 (b) 196 (c) 130 (d) 262
21. 21.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 3)(x + 4) (c) (x + 1)(x + 12) (d) (x + 1)(x + 6)
22. 22.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 7)(x + 8) (c) (x + 1)(x + 14) (d) (x + 1)(x + 56)
23. 23.If a + b = 5 and ab = 4, what is a² + b²? (a) 17 (b) 25 (c) 21 (d) 33
24. 24.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2525 (b) 2025 (c) 3025 (d) 3035
25. 25.If a + b = 6 and ab = 5, what is a² + b²? (a) 36 (b) 46 (c) 31 (d) 26
26. 26.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x − 4)(x − 8) (c) (x + 1)(x + 11) (d) (x + 4)(x + 8)
27. 27.If a + b = 17 and ab = 72, what is (a − b)²? (a) 289 (b) 1 (c) 145 (d) 217
28. 28.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 841 (b) 961 (c) 901 (d) 971
29. 29.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 8084 (b) 7396 (c) 7406 (d) 8836
30. 30.What is the sum of the roots of x² − 7x + 10 = 0? (a) 10 (b) 6 (c) 7 (d) 8
31. 31.If a + b = 15 and ab = 54, what is (a − b)²? (a) 117 (b) 3 (c) 225 (d) 9
32. 32.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x + 2)(x + 4) (c) (x + 1)(x + 8) (d) (x − 2)(x − 4)
33. 33.If a + b = 17 and ab = 70, what is a² + b²? (a) 429 (b) 289 (c) 219 (d) 149
34. 34.What is the sum of the roots of x² − 12x + 35 = 0? (a) 11 (b) 12 (c) 35 (d) 13
35. 35.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 7579 (b) 8091 (c) 7569 (d) 8649
36. 36.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x − 3)(x − 5) (c) (x + 3)(x + 5) (d) (x + 1)(x + 7)
37. 37.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)
38. 38.What is the product of the roots of x² − 11x + 28 = 0? (a) 27 (b) 29 (c) 11 (d) 28
39. 39.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3481 (b) 3601 (c) 3731 (d) 3721
40. 40.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 5329 (b) 4909 (c) 5339 (d) 4489

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