Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 748 · 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 + 12x + 32? (a) (x + 1)(x + 11) (b) (x + 1)(x + 32) (c) (x − 4)(x − 8) (d) (x + 4)(x + 8)
2. 2.If a + b = 8 and ab = 15, what is a² + b²? (a) 64 (b) 94 (c) 49 (d) 34
3. 3.What is the sum of the roots of x² − 9x + 14 = 0? (a) 14 (b) 10 (c) 8 (d) 9
4. 4.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3259 (b) 3591 (c) 3249 (d) 3969
5. 5.What is the sum of the roots of x² − 9x + 20 = 0? (a) 8 (b) 10 (c) 20 (d) 9
6. 6.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x + 1)(x + 24) (c) (x + 3)(x + 8) (d) (x − 3)(x − 8)
7. 7.What is the product of the roots of x² − 10x + 21 = 0? (a) 20 (b) 21 (c) 10 (d) 22
8. 8.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)
9. 9.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 73 (c) 17 (d) 72
10. 10.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 361 (b) 399 (c) 441 (d) 371
11. 11.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2516 (b) 2116 (c) 2926 (d) 2916
12. 12.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)
13. 13.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x − 4)(x − 6) (c) (x + 4)(x + 6) (d) (x + 1)(x + 9)
14. 14.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4356 (b) 4884 (c) 5476 (d) 4366
15. 15.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 2 and 36 (c) 8 and 9 (d) 1 and 72
16. 16.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 202 (c) 4 (d) 400
17. 17.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)
18. 18.If a + b = 14 and ab = 40, what is (a − b)²? (a) 6 (b) 196 (c) 36 (d) 116
19. 19.If a + b = 7 and ab = 10, what is (a − b)²? (a) 29 (b) 49 (c) 9 (d) 3
20. 20.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 7569 (b) 8659 (c) 8649 (d) 8109
21. 21.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 1)(x + 20) (c) (x + 1)(x + 8) (d) (x + 4)(x + 5)
22. 22.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 729 (b) 1099 (c) 1089 (d) 909
23. 23.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 2)(x + 4) (c) (x − 2)(x − 4) (d) (x + 1)(x + 5)
24. 24.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)
25. 25.If a + b = 17 and ab = 70, what is (a − b)²? (a) 149 (b) 289 (c) 3 (d) 9
26. 26.If a + b = 19 and ab = 88, what is (a − b)²? (a) 361 (b) 185 (c) 9 (d) 3
27. 27.What are the roots of x² − 15x + 56 = 0? (a) 2 and 28 (b) 7 and 8 (c) 1 and 14 (d) 1 and 56
28. 28.If a + b = 11 and ab = 10, what is (a − b)²? (a) 9 (b) 101 (c) 81 (d) 121
29. 29.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x − 2)(x − 6) (c) (x + 1)(x + 12) (d) (x + 2)(x + 6)
30. 30.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x + 2)(x + 8) (c) (x − 2)(x − 8) (d) (x + 1)(x + 16)
31. 31.If a + b = 7 and ab = 12, what is (a − b)²? (a) 37 (b) 49 (c) 1 (d) 25
32. 32.What is the product of the roots of x² − 6x + 8 = 0? (a) 7 (b) 9 (c) 6 (d) 8
33. 33.What is the product of the roots of x² − 13x + 42 = 0? (a) 41 (b) 13 (c) 42 (d) 43
34. 34.What are the roots of x² − 8x + 12 = 0? (a) 3 and 4 (b) 2 and 6 (c) 1 and 12 (d) 1 and 7
35. 35.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7235 (b) 7225 (c) 6425 (d) 5625
36. 36.What is the sum of the roots of x² − 15x + 54 = 0? (a) 16 (b) 54 (c) 15 (d) 14
37. 37.What is the sum of the roots of x² − 14x + 48 = 0? (a) 15 (b) 48 (c) 13 (d) 14
38. 38.If a + b = 5 and ab = 4, what is a² + b²? (a) 25 (b) 33 (c) 21 (d) 17
39. 39.If a + b = 13 and ab = 42, what is a² + b²? (a) 85 (b) 169 (c) 253 (d) 127
40. 40.Which is the correct factorisation of x² + 7x + 10? (a) (x + 2)(x + 5) (b) (x + 1)(x + 10) (c) (x + 1)(x + 6) (d) (x − 2)(x − 5)

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