Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 362 · 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.If a + b = 18 and ab = 77, what is a² + b²? (a) 478 (b) 247 (c) 324 (d) 170
3. 3.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 8075 (b) 9025 (c) 7225 (d) 7235
4. 4.If a + b = 12 and ab = 35, what is a² + b²? (a) 74 (b) 214 (c) 109 (d) 144
5. 5.What is the sum of the roots of x² − 11x + 30 = 0? (a) 30 (b) 11 (c) 10 (d) 12
6. 6.If a + b = 8 and ab = 7, what is a² + b²? (a) 50 (b) 64 (c) 57 (d) 78
7. 7.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 6) (c) (x + 3)(x + 4) (d) (x + 1)(x + 12)
8. 8.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1936 (b) 1616 (c) 1296 (d) 1946
9. 9.What is the sum of the roots of x² − 11x + 24 = 0? (a) 24 (b) 10 (c) 11 (d) 12
10. 10.What are the roots of x² − 6x + 8 = 0? (a) 1 and 8 (b) 2 and 5 (c) 1 and 5 (d) 2 and 4
11. 11.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x − 6)(x − 8) (c) (x + 1)(x + 48) (d) (x + 6)(x + 8)
12. 12.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 72 (c) 18 (d) 17
13. 13.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 15 (c) 56 (d) 55
14. 14.What is the product of the roots of x² − 15x + 54 = 0? (a) 53 (b) 15 (c) 55 (d) 54
15. 15.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3599 (b) 3721 (c) 3481 (d) 3491
16. 16.If a + b = 14 and ab = 40, what is a² + b²? (a) 276 (b) 196 (c) 156 (d) 116
17. 17.If a + b = 12 and ab = 20, what is a² + b²? (a) 104 (b) 124 (c) 184 (d) 144
18. 18.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2304 (b) 2504 (c) 2714 (d) 2704
19. 19.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 4 (c) 400 (d) 202
20. 20.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 7)(x + 8) (c) (x + 1)(x + 56) (d) (x − 7)(x − 8)
21. 21.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4925 (b) 5635 (c) 4225 (d) 5625
22. 22.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10201 (b) 10001 (c) 10211 (d) 9801
23. 23.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)
24. 24.If a + b = 12 and ab = 32, what is a² + b²? (a) 208 (b) 112 (c) 144 (d) 80
25. 25.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 541 (c) 181 (d) 271
26. 26.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 324 (b) 334 (c) 396 (d) 484
27. 27.If a + b = 21 and ab = 108, what is (a − b)²? (a) 3 (b) 441 (c) 9 (d) 225
28. 28.If a + b = 19 and ab = 88, what is (a − b)²? (a) 9 (b) 185 (c) 361 (d) 3
29. 29.What is the sum of the roots of x² − 11x + 18 = 0? (a) 18 (b) 10 (c) 12 (d) 11
30. 30.If a + b = 15 and ab = 56, what is a² + b²? (a) 113 (b) 337 (c) 169 (d) 225
31. 31.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 5184 (b) 4896 (c) 4634 (d) 4624
32. 32.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x + 2)(x + 5) (c) (x − 2)(x − 5) (d) (x + 1)(x + 6)
33. 33.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6571 (b) 6561 (c) 6401 (d) 6241
34. 34.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1099 (b) 729 (c) 1089 (d) 909
35. 35.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 1 and 16 (c) 8 and 9 (d) 2 and 36
36. 36.If a + b = 16 and ab = 55, what is a² + b²? (a) 256 (b) 201 (c) 146 (d) 366
37. 37.If a + b = 17 and ab = 70, what is (a − b)²? (a) 149 (b) 9 (c) 3 (d) 289
38. 38.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 3)(x + 6) (c) (x + 1)(x + 8) (d) (x + 1)(x + 18)
39. 39.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)
40. 40.What are the roots of x² − 10x + 24 = 0? (a) 2 and 12 (b) 4 and 6 (c) 1 and 9 (d) 1 and 24

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