Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 858 · 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.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 15 (c) 63 (d) 16
2. 2.What is the sum of the roots of x² − 13x + 42 = 0? (a) 12 (b) 14 (c) 42 (d) 13
3. 3.If a + b = 16 and ab = 55, what is (a − b)²? (a) 36 (b) 256 (c) 6 (d) 146
4. 4.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 15) (c) (x − 7)(x − 9) (d) (x + 1)(x + 63)
5. 5.Which is the correct factorisation of x² + 11x + 28? (a) (x + 4)(x + 7) (b) (x − 4)(x − 7) (c) (x + 1)(x + 10) (d) (x + 1)(x + 28)
6. 6.If a + b = 6 and ab = 8, what is (a − b)²? (a) 20 (b) 2 (c) 36 (d) 4
7. 7.What is the product of the roots of x² − 11x + 18 = 0? (a) 17 (b) 18 (c) 19 (d) 11
8. 8.If a + b = 12 and ab = 27, what is a² + b²? (a) 90 (b) 117 (c) 198 (d) 144
9. 9.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 225 (c) 9 (d) 3
10. 10.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 7579 (b) 8091 (c) 7569 (d) 8649
11. 11.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6399 (b) 6251 (c) 6241 (d) 6561
12. 12.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3721 (b) 3731 (c) 3601 (d) 3481
13. 13.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1596 (b) 1764 (c) 1454 (d) 1444
14. 14.What is the product of the roots of x² − 11x + 24 = 0? (a) 23 (b) 24 (c) 11 (d) 25
15. 15.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 7)(x + 8) (c) (x − 7)(x − 8) (d) (x + 1)(x + 56)
16. 16.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1604 (b) 1444 (c) 1774 (d) 1764
17. 17.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
18. 18.If a + b = 19 and ab = 90, what is a² + b²? (a) 181 (b) 541 (c) 271 (d) 361
19. 19.If a + b = 16 and ab = 48, what is (a − b)²? (a) 256 (b) 64 (c) 160 (d) 8
20. 20.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 71 (c) 17 (d) 73
21. 21.What are the roots of x² − 13x + 42 = 0? (a) 6 and 7 (b) 2 and 21 (c) 1 and 42 (d) 1 and 12
22. 22.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3969 (b) 3979 (c) 3609 (d) 3249
23. 23.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 4)(x + 5) (c) (x + 1)(x + 8) (d) (x + 1)(x + 20)
24. 24.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 676 (b) 686 (c) 1156 (d) 884
25. 25.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x + 1)(x + 9) (c) (x − 4)(x − 6) (d) (x + 4)(x + 6)
26. 26.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1024 (b) 784 (c) 1034 (d) 904
27. 27.What is the product of the roots of x² − 6x + 8 = 0? (a) 6 (b) 7 (c) 9 (d) 8
28. 28.What is the product of the roots of x² − 16x + 63 = 0? (a) 62 (b) 63 (c) 16 (d) 64
29. 29.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2401 (b) 2411 (c) 2499 (d) 2601
30. 30.Which is the correct factorisation of x² + 8x + 15? (a) (x + 3)(x + 5) (b) (x − 3)(x − 5) (c) (x + 1)(x + 7) (d) (x + 1)(x + 15)
31. 31.If a + b = 10 and ab = 24, what is (a − b)²? (a) 100 (b) 52 (c) 4 (d) 2
32. 32.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 1)(x + 35) (c) (x + 5)(x + 7) (d) (x + 1)(x + 11)
33. 33.What is the product of the roots of x² − 11x + 30 = 0? (a) 11 (b) 31 (c) 30 (d) 29
34. 34.What are the roots of x² − 15x + 56 = 0? (a) 7 and 8 (b) 1 and 14 (c) 1 and 56 (d) 2 and 28
35. 35.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x + 2)(x + 5) (c) (x + 1)(x + 10) (d) (x − 2)(x − 5)
36. 36.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 916 (b) 1166 (c) 1156 (d) 676
37. 37.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)
38. 38.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4925 (b) 4225 (c) 5625 (d) 5635
39. 39.If a + b = 5 and ab = 6, what is (a − b)²? (a) 13 (b) 25 (c) 1 (d) 19
40. 40.If a + b = 14 and ab = 33, what is a² + b²? (a) 196 (b) 262 (c) 163 (d) 130

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