Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 720 · 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.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2819 (b) 2509 (c) 2809 (d) 2209
2. 2.What are the roots of x² − 15x + 56 = 0? (a) 1 and 14 (b) 7 and 8 (c) 2 and 28 (d) 1 and 56
3. 3.If a + b = 17 and ab = 66, what is a² + b²? (a) 289 (b) 157 (c) 223 (d) 421
4. 4.If a + b = 17 and ab = 60, what is (a − b)²? (a) 289 (b) 7 (c) 49 (d) 169
5. 5.If a + b = 15 and ab = 44, what is (a − b)²? (a) 7 (b) 225 (c) 49 (d) 137
6. 6.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 1)(x + 11) (c) (x + 1)(x + 27) (d) (x + 3)(x + 9)
7. 7.What is the product of the roots of x² − 11x + 18 = 0? (a) 11 (b) 18 (c) 17 (d) 19
8. 8.If a + b = 11 and ab = 18, what is (a − b)²? (a) 121 (b) 85 (c) 49 (d) 7
9. 9.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 11025 (b) 9025 (c) 11035 (d) 10025
10. 10.Which is the correct factorisation of x² + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 1)(x + 16) (c) (x + 1)(x + 9) (d) (x + 2)(x + 8)
11. 11.What is the product of the roots of x² − 7x + 10 = 0? (a) 7 (b) 9 (c) 10 (d) 11
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 + 45) (d) (x + 1)(x + 13)
13. 13.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
14. 14.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 961 (b) 901 (c) 971 (d) 841
15. 15.Which is the correct factorisation of x² + 11x + 24? (a) (x − 3)(x − 8) (b) (x + 1)(x + 10) (c) (x + 1)(x + 24) (d) (x + 3)(x + 8)
16. 16.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x + 5)(x + 8) (c) (x + 1)(x + 40) (d) (x − 5)(x − 8)
17. 17.What is the sum of the roots of x² − 9x + 20 = 0? (a) 20 (b) 10 (c) 8 (d) 9
18. 18.If a + b = 17 and ab = 60, what is a² + b²? (a) 409 (b) 169 (c) 289 (d) 229
19. 19.Which is the correct factorisation of x² + 9x + 18? (a) (x + 3)(x + 6) (b) (x − 3)(x − 6) (c) (x + 1)(x + 18) (d) (x + 1)(x + 8)
20. 20.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 16 (c) 17 (d) 72
21. 21.What is the product of the roots of x² − 11x + 30 = 0? (a) 29 (b) 31 (c) 11 (d) 30
22. 22.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4366 (b) 4356 (c) 4884 (d) 5476
23. 23.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 1)(x + 10) (c) (x + 1)(x + 30) (d) (x + 5)(x + 6)
24. 24.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x − 3)(x − 5) (c) (x + 3)(x + 5) (d) (x + 1)(x + 7)
25. 25.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x − 2)(x − 5) (c) (x + 2)(x + 5) (d) (x + 1)(x + 10)
26. 26.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6889 (b) 6391 (c) 5939 (d) 5929
27. 27.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 5)(x + 7) (c) (x + 1)(x + 11) (d) (x + 1)(x + 35)
28. 28.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x + 7)(x + 8) (d) (x − 7)(x − 8)
29. 29.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 396 (b) 334 (c) 484 (d) 324
30. 30.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9996 (b) 9614 (c) 9604 (d) 10404
31. 31.If a + b = 10 and ab = 16, what is (a − b)²? (a) 100 (b) 68 (c) 36 (d) 6
32. 32.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3721 (b) 3481 (c) 3601 (d) 3731
33. 33.If a + b = 19 and ab = 90, what is a² + b²? (a) 181 (b) 271 (c) 361 (d) 541
34. 34.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9975 (b) 9035 (c) 9025 (d) 11025
35. 35.If a + b = 7 and ab = 10, what is a² + b²? (a) 69 (b) 39 (c) 49 (d) 29
36. 36.If a + b = 17 and ab = 70, what is a² + b²? (a) 289 (b) 219 (c) 149 (d) 429
37. 37.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2504 (b) 2714 (c) 2704 (d) 2304
38. 38.If a + b = 10 and ab = 24, what is (a − b)²? (a) 2 (b) 100 (c) 4 (d) 52
39. 39.If a + b = 20 and ab = 99, what is a² + b²? (a) 400 (b) 301 (c) 202 (d) 598
40. 40.What is the product of the roots of x² − 10x + 21 = 0? (a) 10 (b) 20 (c) 22 (d) 21

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