Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 888 · 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.Which is the correct factorisation of x² + 8x + 12? (a) (x + 2)(x + 6) (b) (x − 2)(x − 6) (c) (x + 1)(x + 12) (d) (x + 1)(x + 7)
2. 2.If a + b = 17 and ab = 70, what is (a − b)²? (a) 149 (b) 289 (c) 9 (d) 3
3. 3.If a + b = 16 and ab = 63, what is (a − b)²? (a) 4 (b) 2 (c) 256 (d) 130
4. 4.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 18) (b) (x − 2)(x − 9) (c) (x + 1)(x + 10) (d) (x + 2)(x + 9)
5. 5.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4489 (b) 4891 (c) 4499 (d) 5329
6. 6.If a + b = 12 and ab = 20, what is (a − b)²? (a) 104 (b) 64 (c) 144 (d) 8
7. 7.If a + b = 5 and ab = 4, what is (a − b)²? (a) 3 (b) 17 (c) 25 (d) 9
8. 8.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x + 1)(x + 40) (c) (x − 5)(x − 8) (d) (x + 5)(x + 8)
9. 9.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x + 4)(x + 6) (c) (x − 4)(x − 6) (d) (x + 1)(x + 9)
10. 10.If a + b = 6 and ab = 5, what is (a − b)²? (a) 26 (b) 16 (c) 4 (d) 36
11. 11.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 2)(x + 7) (c) (x + 1)(x + 14) (d) (x + 1)(x + 8)
12. 12.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8291 (b) 8101 (c) 8281 (d) 7921
13. 13.If a + b = 8 and ab = 7, what is (a − b)²? (a) 36 (b) 64 (c) 6 (d) 50
14. 14.What are the roots of x² − 13x + 42 = 0? (a) 1 and 12 (b) 2 and 21 (c) 1 and 42 (d) 6 and 7
15. 15.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 5184 (b) 4624 (c) 4904 (d) 5194
16. 16.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x + 1)(x + 16) (c) (x − 2)(x − 8) (d) (x + 2)(x + 8)
17. 17.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 256 (b) 266 (c) 384 (d) 576
18. 18.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 4225 (b) 3025 (c) 4235 (d) 3625
19. 19.Which is the correct factorisation of x² + 5x + 6? (a) (x + 2)(x + 3) (b) (x + 1)(x + 6) (c) (x + 1)(x + 4) (d) (x − 2)(x − 3)
20. 20.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1521 (b) 1601 (c) 1681 (d) 1691
21. 21.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 441 (c) 9 (d) 3
22. 22.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)
23. 23.If a + b = 6 and ab = 8, what is (a − b)²? (a) 20 (b) 36 (c) 2 (d) 4
24. 24.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1625 (b) 1225 (c) 2025 (d) 2035
25. 25.What is the product of the roots of x² − 7x + 10 = 0? (a) 10 (b) 11 (c) 9 (d) 7
26. 26.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4225 (b) 5635 (c) 4925 (d) 5625
27. 27.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3599 (b) 3491 (c) 3721 (d) 3481
28. 28.If a + b = 9 and ab = 18, what is a² + b²? (a) 117 (b) 63 (c) 81 (d) 45
29. 29.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 7931 (b) 8099 (c) 8281 (d) 7921
30. 30.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 9025 (b) 11035 (c) 11025 (d) 10025
31. 31.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 1156 (b) 676 (c) 884 (d) 686
32. 32.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 961 (b) 841 (c) 901 (d) 971
33. 33.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 5625 (b) 6425 (c) 7225 (d) 7235
34. 34.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x − 7)(x − 8) (c) (x + 1)(x + 14) (d) (x + 7)(x + 8)
35. 35.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 4)(x + 9) (c) (x + 1)(x + 12) (d) (x + 1)(x + 36)
36. 36.If a + b = 13 and ab = 40, what is a² + b²? (a) 129 (b) 249 (c) 169 (d) 89
37. 37.What are the roots of x² − 8x + 15 = 0? (a) 3 and 5 (b) 2 and 6 (c) 1 and 15 (d) 1 and 7
38. 38.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)
39. 39.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1156 (b) 676 (c) 1166 (d) 916
40. 40.What is the sum of the roots of x² − 8x + 15 = 0? (a) 8 (b) 7 (c) 9 (d) 15

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