Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 717 · 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 = 14 and ab = 33, what is a² + b²? (a) 262 (b) 196 (c) 130 (d) 163
2. 2.If a + b = 7 and ab = 10, what is a² + b²? (a) 39 (b) 69 (c) 49 (d) 29
3. 3.What are the roots of x² − 7x + 12 = 0? (a) 3 and 4 (b) 2 and 6 (c) 1 and 6 (d) 1 and 12
4. 4.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 5184 (b) 4624 (c) 4634 (d) 4896
5. 5.What is the sum of the roots of x² − 11x + 18 = 0? (a) 18 (b) 12 (c) 11 (d) 10
6. 6.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 17 (d) 16
7. 7.If a + b = 15 and ab = 54, what is a² + b²? (a) 333 (b) 225 (c) 171 (d) 117
8. 8.If a + b = 15 and ab = 56, what is (a − b)²? (a) 1 (b) 225 (c) 113 (d) 169
9. 9.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 8084 (b) 7396 (c) 7406 (d) 8836
10. 10.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 2025 (b) 1235 (c) 1225 (d) 1575
11. 11.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x + 3)(x + 6) (c) (x − 3)(x − 6) (d) (x + 1)(x + 8)
12. 12.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 5)(x + 9) (c) (x + 1)(x + 13) (d) (x + 1)(x + 45)
13. 13.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1099 (b) 909 (c) 729 (d) 1089
14. 14.If a + b = 13 and ab = 30, what is (a − b)²? (a) 109 (b) 7 (c) 49 (d) 169
15. 15.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 4 (c) 400 (d) 202
16. 16.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x + 1)(x + 12) (c) (x + 2)(x + 6) (d) (x − 2)(x − 6)
17. 17.If a + b = 15 and ab = 44, what is a² + b²? (a) 137 (b) 225 (c) 181 (d) 313
18. 18.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x − 6)(x − 7) (c) (x + 6)(x + 7) (d) (x + 1)(x + 12)
19. 19.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 971 (b) 961 (c) 841 (d) 901
20. 20.What is the product of the roots of x² − 10x + 24 = 0? (a) 25 (b) 24 (c) 23 (d) 10
21. 21.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 1)(x + 15) (c) (x − 7)(x − 9) (d) (x + 7)(x + 9)
22. 22.If a + b = 13 and ab = 42, what is (a − b)²? (a) 1 (b) 127 (c) 169 (d) 85
23. 23.What are the roots of x² − 8x + 12 = 0? (a) 3 and 4 (b) 1 and 12 (c) 1 and 7 (d) 2 and 6
24. 24.If a + b = 19 and ab = 88, what is (a − b)²? (a) 9 (b) 361 (c) 185 (d) 3
25. 25.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6251 (b) 6399 (c) 6241 (d) 6561
26. 26.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
27. 27.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7744 (b) 8096 (c) 8464 (d) 7754
28. 28.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10211 (b) 10201 (c) 9801 (d) 10001
29. 29.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x − 6)(x − 9) (c) (x + 1)(x + 54) (d) (x + 1)(x + 14)
30. 30.If a + b = 8 and ab = 12, what is a² + b²? (a) 64 (b) 40 (c) 88 (d) 52
31. 31.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4356 (b) 5476 (c) 4916 (d) 5486
32. 32.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x + 4)(x + 7) (c) (x + 1)(x + 28) (d) (x − 4)(x − 7)
33. 33.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 586 (c) 416 (d) 576
34. 34.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4904 (b) 4624 (c) 5194 (d) 5184
35. 35.What are the roots of x² − 16x + 63 = 0? (a) 3 and 21 (b) 1 and 63 (c) 1 and 15 (d) 7 and 9
36. 36.What is the product of the roots of x² − 16x + 63 = 0? (a) 16 (b) 63 (c) 62 (d) 64
37. 37.If a + b = 9 and ab = 14, what is a² + b²? (a) 109 (b) 81 (c) 67 (d) 53
38. 38.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3259 (b) 3591 (c) 3969 (d) 3249
39. 39.What is the product of the roots of x² − 10x + 21 = 0? (a) 20 (b) 10 (c) 21 (d) 22
40. 40.What are the roots of x² − 11x + 30 = 0? (a) 5 and 6 (b) 1 and 30 (c) 2 and 15 (d) 1 and 10

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