Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 372 · 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 = 12 and ab = 32, what is (a − b)²? (a) 4 (b) 144 (c) 80 (d) 16
2. 2.If a + b = 12 and ab = 27, what is (a − b)²? (a) 90 (b) 144 (c) 36 (d) 6
3. 3.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10001 (b) 10211 (c) 10201 (d) 9801
4. 4.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 16 (c) 17 (d) 18
5. 5.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2401 (b) 2611 (c) 2601 (d) 2501
6. 6.If a + b = 17 and ab = 60, what is (a − b)²? (a) 169 (b) 289 (c) 7 (d) 49
7. 7.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 2025 (b) 1235 (c) 1575 (d) 1225
8. 8.If a + b = 12 and ab = 35, what is (a − b)²? (a) 74 (b) 144 (c) 2 (d) 4
9. 9.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4356 (b) 5476 (c) 4366 (d) 4884
10. 10.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1625 (b) 2025 (c) 2035 (d) 1225
11. 11.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 12) (c) (x + 6)(x + 7) (d) (x + 1)(x + 42)
12. 12.If a + b = 9 and ab = 8, what is (a − b)²? (a) 81 (b) 7 (c) 49 (d) 65
13. 13.What is the product of the roots of x² − 13x + 42 = 0? (a) 13 (b) 42 (c) 43 (d) 41
14. 14.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 71 (c) 17 (d) 73
15. 15.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 9025 (b) 10025 (c) 11035 (d) 11025
16. 16.If a + b = 16 and ab = 55, what is a² + b²? (a) 256 (b) 366 (c) 201 (d) 146
17. 17.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x + 1)(x + 15) (c) (x + 3)(x + 5) (d) (x − 3)(x − 5)
18. 18.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 5194 (b) 4904 (c) 5184 (d) 4624
19. 19.If a + b = 19 and ab = 88, what is a² + b²? (a) 361 (b) 537 (c) 185 (d) 273
20. 20.If a + b = 17 and ab = 66, what is a² + b²? (a) 223 (b) 421 (c) 289 (d) 157
21. 21.What is the product of the roots of x² − 12x + 35 = 0? (a) 35 (b) 34 (c) 12 (d) 36
22. 22.If a + b = 15 and ab = 50, what is (a − b)²? (a) 125 (b) 25 (c) 225 (d) 5
23. 23.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 8075 (b) 7225 (c) 9025 (d) 7235
24. 24.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)
25. 25.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)
26. 26.What are the roots of x² − 12x + 32 = 0? (a) 4 and 8 (b) 2 and 16 (c) 1 and 32 (d) 1 and 11
27. 27.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 1)(x + 13) (c) (x + 1)(x + 45) (d) (x + 5)(x + 9)
28. 28.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3854 (b) 3604 (c) 3844 (d) 3364
29. 29.Which is the correct factorisation of x² + 10x + 24? (a) (x + 4)(x + 6) (b) (x + 1)(x + 9) (c) (x − 4)(x − 6) (d) (x + 1)(x + 24)
30. 30.Which is the correct factorisation of x² + 10x + 21? (a) (x + 3)(x + 7) (b) (x + 1)(x + 21) (c) (x + 1)(x + 9) (d) (x − 3)(x − 7)
31. 31.If a + b = 15 and ab = 36, what is a² + b²? (a) 189 (b) 225 (c) 153 (d) 297
32. 32.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3616 (b) 4106 (c) 3136 (d) 4096
33. 33.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 56 (c) 14 (d) 16
34. 34.What is the sum of the roots of x² − 6x + 8 = 0? (a) 7 (b) 8 (c) 5 (d) 6
35. 35.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 1)(x + 11) (c) (x + 5)(x + 7) (d) (x + 1)(x + 35)
36. 36.If a + b = 5 and ab = 6, what is a² + b²? (a) 37 (b) 25 (c) 19 (d) 13
37. 37.What are the roots of x² − 11x + 30 = 0? (a) 1 and 30 (b) 2 and 15 (c) 1 and 10 (d) 5 and 6
38. 38.If a + b = 14 and ab = 48, what is a² + b²? (a) 292 (b) 196 (c) 100 (d) 148
39. 39.What is the sum of the roots of x² − 12x + 32 = 0? (a) 11 (b) 32 (c) 13 (d) 12
40. 40.What is the product of the roots of x² − 10x + 24 = 0? (a) 23 (b) 24 (c) 25 (d) 10

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