Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 33 · 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 + 17x + 72? (a) (x + 8)(x + 9) (b) (x − 8)(x − 9) (c) (x + 1)(x + 72) (d) (x + 1)(x + 16)
2. 2.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2501 (b) 2601 (c) 2611 (d) 2401
3. 3.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9811 (b) 9801 (c) 9999 (d) 10201
4. 4.What is the product of the roots of x² − 13x + 36 = 0? (a) 13 (b) 35 (c) 36 (d) 37
5. 5.What are the roots of x² − 10x + 21 = 0? (a) 3 and 7 (b) 1 and 9 (c) 1 and 21 (d) 2 and 8
6. 6.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 1)(x + 45) (c) (x + 5)(x + 9) (d) (x + 1)(x + 13)
7. 7.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x + 1)(x + 10) (c) (x − 2)(x − 5) (d) (x + 2)(x + 5)
8. 8.If a + b = 13 and ab = 30, what is a² + b²? (a) 229 (b) 109 (c) 169 (d) 139
9. 9.Which is the correct factorisation of x² + 13x + 36? (a) (x + 4)(x + 9) (b) (x + 1)(x + 12) (c) (x + 1)(x + 36) (d) (x − 4)(x − 9)
10. 10.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7066 (b) 6416 (c) 7056 (d) 5776
11. 11.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 2 and 28 (c) 7 and 8 (d) 1 and 14
12. 12.If a + b = 6 and ab = 5, what is a² + b²? (a) 36 (b) 46 (c) 26 (d) 31
13. 13.What is the sum of the roots of x² − 7x + 12 = 0? (a) 6 (b) 12 (c) 8 (d) 7
14. 14.What is the sum of the roots of x² − 13x + 40 = 0? (a) 40 (b) 12 (c) 13 (d) 14
15. 15.If a + b = 11 and ab = 18, what is a² + b²? (a) 157 (b) 121 (c) 103 (d) 85
16. 16.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 384 (b) 256 (c) 576 (d) 266
17. 17.If a + b = 17 and ab = 70, what is a² + b²? (a) 149 (b) 289 (c) 219 (d) 429
18. 18.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 56 (c) 15 (d) 14
19. 19.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)
20. 20.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1521 (b) 1531 (c) 1599 (d) 1681
21. 21.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 7225 (b) 5625 (c) 6375 (d) 5635
22. 22.If a + b = 8 and ab = 15, what is a² + b²? (a) 64 (b) 49 (c) 34 (d) 94
23. 23.What are the roots of x² − 10x + 24 = 0? (a) 1 and 9 (b) 4 and 6 (c) 1 and 24 (d) 2 and 12
24. 24.Which is the correct factorisation of x² + 10x + 21? (a) (x − 3)(x − 7) (b) (x + 3)(x + 7) (c) (x + 1)(x + 21) (d) (x + 1)(x + 9)
25. 25.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4356 (b) 5476 (c) 4916 (d) 5486
26. 26.If a + b = 12 and ab = 11, what is (a − b)²? (a) 122 (b) 144 (c) 10 (d) 100
27. 27.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x − 3)(x − 5) (c) (x + 1)(x + 15) (d) (x + 3)(x + 5)
28. 28.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 7)(x + 9) (c) (x + 1)(x + 63) (d) (x + 1)(x + 15)
29. 29.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 7579 (b) 8091 (c) 8649 (d) 7569
30. 30.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 8075 (b) 7225 (c) 9025 (d) 7235
31. 31.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3481 (b) 3721 (c) 3731 (d) 3601
32. 32.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6251 (b) 6241 (c) 6561 (d) 6399
33. 33.If a + b = 9 and ab = 14, what is (a − b)²? (a) 53 (b) 25 (c) 81 (d) 5
34. 34.What is the sum of the roots of x² − 11x + 30 = 0? (a) 10 (b) 30 (c) 12 (d) 11
35. 35.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 235 (b) 625 (c) 375 (d) 225
36. 36.What is the product of the roots of x² − 12x + 32 = 0? (a) 33 (b) 12 (c) 32 (d) 31
37. 37.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x − 3)(x − 4) (c) (x + 3)(x + 4) (d) (x + 1)(x + 6)
38. 38.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9614 (b) 9604 (c) 10404 (d) 9996
39. 39.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)
40. 40.What are the roots of x² − 8x + 12 = 0? (a) 3 and 4 (b) 1 and 7 (c) 1 and 12 (d) 2 and 6

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