Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 771 · 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 = 8 and ab = 12, what is (a − b)²? (a) 64 (b) 16 (c) 40 (d) 4
2. 2.If a + b = 9 and ab = 14, what is a² + b²? (a) 81 (b) 53 (c) 67 (d) 109
3. 3.What are the roots of x² − 12x + 27 = 0? (a) 1 and 27 (b) 1 and 11 (c) 3 and 9 (d) 2 and 10
4. 4.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 16 (c) 63 (d) 15
5. 5.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3604 (b) 3854 (c) 3844 (d) 3364
6. 6.What are the roots of x² − 11x + 24 = 0? (a) 1 and 10 (b) 1 and 24 (c) 2 and 12 (d) 3 and 8
7. 7.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 15 (c) 14 (d) 13
8. 8.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1936 (b) 1946 (c) 1616 (d) 1296
9. 9.What is the sum of the roots of x² − 14x + 45 = 0? (a) 15 (b) 45 (c) 13 (d) 14
10. 10.What are the roots of x² − 9x + 20 = 0? (a) 1 and 8 (b) 4 and 5 (c) 2 and 10 (d) 1 and 20
11. 11.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 14) (c) (x + 1)(x + 54) (d) (x + 6)(x + 9)
12. 12.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x + 3)(x + 8) (c) (x − 3)(x − 8) (d) (x + 1)(x + 24)
13. 13.If a + b = 13 and ab = 36, what is a² + b²? (a) 169 (b) 97 (c) 133 (d) 241
14. 14.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 2 and 36 (c) 8 and 9 (d) 1 and 16
15. 15.What are the roots of x² − 16x + 63 = 0? (a) 3 and 21 (b) 7 and 9 (c) 1 and 15 (d) 1 and 63
16. 16.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 576 (b) 384 (c) 266 (d) 256
17. 17.If a + b = 16 and ab = 63, what is (a − b)²? (a) 4 (b) 256 (c) 2 (d) 130
18. 18.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8109 (b) 8659 (c) 7569 (d) 8649
19. 19.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10414 (b) 10404 (c) 9604 (d) 10004
20. 20.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 12) (c) (x + 1)(x + 42) (d) (x + 6)(x + 7)
21. 21.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 1)(x + 63) (c) (x + 7)(x + 9) (d) (x − 7)(x − 9)
22. 22.Which is the correct factorisation of x² + 7x + 12? (a) (x + 3)(x + 4) (b) (x − 3)(x − 4) (c) (x + 1)(x + 6) (d) (x + 1)(x + 12)
23. 23.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1306 (b) 1296 (c) 1584 (d) 1936
24. 24.What is the product of the roots of x² − 11x + 24 = 0? (a) 23 (b) 25 (c) 11 (d) 24
25. 25.What is the product of the roots of x² − 11x + 30 = 0? (a) 30 (b) 29 (c) 31 (d) 11
26. 26.If a + b = 13 and ab = 36, what is (a − b)²? (a) 97 (b) 25 (c) 5 (d) 169
27. 27.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10609 (b) 10009 (c) 9409 (d) 10619
28. 28.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x + 1)(x + 8) (c) (x + 2)(x + 7) (d) (x − 2)(x − 7)
29. 29.What is the sum of the roots of x² − 5x + 6 = 0? (a) 5 (b) 6 (c) 7 (d) 4
30. 30.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 72 (c) 73 (d) 17
31. 31.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2484 (b) 2116 (c) 2126 (d) 2916
32. 32.If a + b = 9 and ab = 20, what is a² + b²? (a) 81 (b) 121 (c) 61 (d) 41
33. 33.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x + 3)(x + 5) (c) (x + 1)(x + 7) (d) (x − 3)(x − 5)
34. 34.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 1)(x + 27) (c) (x + 3)(x + 9) (d) (x + 1)(x + 11)
35. 35.What are the roots of x² − 10x + 24 = 0? (a) 4 and 6 (b) 1 and 24 (c) 2 and 12 (d) 1 and 9
36. 36.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x − 8)(x − 9) (c) (x + 8)(x + 9) (d) (x + 1)(x + 16)
37. 37.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 961 (b) 901 (c) 841 (d) 971
38. 38.What is the sum of the roots of x² − 13x + 42 = 0? (a) 13 (b) 12 (c) 42 (d) 14
39. 39.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 484 (b) 404 (c) 324 (d) 494
40. 40.What are the roots of x² − 14x + 48 = 0? (a) 1 and 13 (b) 2 and 24 (c) 1 and 48 (d) 6 and 8

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