Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 104 · 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 + 27? (a) (x + 1)(x + 27) (b) (x − 3)(x − 9) (c) (x + 3)(x + 9) (d) (x + 1)(x + 11)
2. 2.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)
3. 3.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 404 (b) 494 (c) 484 (d) 324
4. 4.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 1089 (b) 739 (c) 891 (d) 729
5. 5.Which is the correct factorisation of x² + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 1)(x + 16) (c) (x + 2)(x + 8) (d) (x + 1)(x + 9)
6. 6.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 1)(x + 10) (c) (x + 5)(x + 6) (d) (x − 5)(x − 6)
7. 7.If a + b = 20 and ab = 96, what is a² + b²? (a) 304 (b) 592 (c) 400 (d) 208
8. 8.What is the product of the roots of x² − 11x + 28 = 0? (a) 28 (b) 11 (c) 27 (d) 29
9. 9.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 16 (c) 56 (d) 15
10. 10.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 961 (b) 841 (c) 899 (d) 851
11. 11.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 7)(x + 9) (c) (x − 7)(x − 9) (d) (x + 1)(x + 15)
12. 12.If a + b = 14 and ab = 33, what is a² + b²? (a) 196 (b) 163 (c) 262 (d) 130
13. 13.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 5635 (b) 4225 (c) 5625 (d) 4925
14. 14.What is the sum of the roots of x² − 5x + 6 = 0? (a) 4 (b) 6 (c) 7 (d) 5
15. 15.Which is the correct factorisation of x² + 13x + 40? (a) (x + 5)(x + 8) (b) (x + 1)(x + 40) (c) (x − 5)(x − 8) (d) (x + 1)(x + 12)
16. 16.If a + b = 19 and ab = 84, what is a² + b²? (a) 361 (b) 193 (c) 529 (d) 277
17. 17.If a + b = 12 and ab = 11, what is (a − b)²? (a) 122 (b) 100 (c) 10 (d) 144
18. 18.What is the sum of the roots of x² − 9x + 20 = 0? (a) 20 (b) 10 (c) 8 (d) 9
19. 19.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 657 (c) 333 (d) 225
20. 20.If a + b = 12 and ab = 27, what is (a − b)²? (a) 144 (b) 90 (c) 6 (d) 36
21. 21.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 334 (b) 396 (c) 324 (d) 484
22. 22.What is the sum of the roots of x² − 7x + 12 = 0? (a) 7 (b) 12 (c) 6 (d) 8
23. 23.What is the sum of the roots of x² − 13x + 40 = 0? (a) 40 (b) 13 (c) 12 (d) 14
24. 24.Which is the correct factorisation of x² + 8x + 15? (a) (x + 3)(x + 5) (b) (x + 1)(x + 15) (c) (x − 3)(x − 5) (d) (x + 1)(x + 7)
25. 25.What is the product of the roots of x² − 9x + 14 = 0? (a) 15 (b) 9 (c) 14 (d) 13
26. 26.What are the roots of x² − 10x + 24 = 0? (a) 4 and 6 (b) 1 and 9 (c) 1 and 24 (d) 2 and 12
27. 27.If a + b = 15 and ab = 56, what is a² + b²? (a) 337 (b) 225 (c) 113 (d) 169
28. 28.If a + b = 13 and ab = 36, what is (a − b)²? (a) 5 (b) 25 (c) 169 (d) 97
29. 29.What is the sum of the roots of x² − 12x + 35 = 0? (a) 12 (b) 35 (c) 13 (d) 11
30. 30.If a + b = 9 and ab = 14, what is a² + b²? (a) 53 (b) 81 (c) 67 (d) 109
31. 31.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 5339 (b) 4489 (c) 5329 (d) 4909
32. 32.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 57 (c) 56 (d) 55
33. 33.If a + b = 20 and ab = 96, what is (a − b)²? (a) 400 (b) 16 (c) 208 (d) 4
34. 34.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 1 and 72 (c) 8 and 9 (d) 2 and 36
35. 35.What are the roots of x² − 14x + 48 = 0? (a) 1 and 48 (b) 1 and 13 (c) 2 and 24 (d) 6 and 8
36. 36.If a + b = 14 and ab = 48, what is a² + b²? (a) 292 (b) 196 (c) 100 (d) 148
37. 37.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)
38. 38.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3259 (b) 3249 (c) 3969 (d) 3591
39. 39.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8109 (b) 7569 (c) 8649 (d) 8659
40. 40.If a + b = 20 and ab = 99, what is a² + b²? (a) 400 (b) 301 (c) 202 (d) 598

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