Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 335 · 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 = 18 and ab = 80, what is (a − b)²? (a) 324 (b) 2 (c) 4 (d) 164
2. 2.What is the sum of the roots of x² − 11x + 18 = 0? (a) 18 (b) 12 (c) 10 (d) 11
3. 3.If a + b = 16 and ab = 60, what is a² + b²? (a) 256 (b) 196 (c) 136 (d) 376
4. 4.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2809 (b) 2509 (c) 2819 (d) 2209
5. 5.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)
6. 6.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
7. 7.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 686 (b) 676 (c) 884 (d) 1156
8. 8.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 6425 (b) 7235 (c) 7225 (d) 5625
9. 9.If a + b = 18 and ab = 72, what is (a − b)²? (a) 324 (b) 180 (c) 6 (d) 36
10. 10.If a + b = 17 and ab = 72, what is a² + b²? (a) 217 (b) 433 (c) 145 (d) 289
11. 11.What is the sum of the roots of x² − 6x + 8 = 0? (a) 8 (b) 5 (c) 6 (d) 7
12. 12.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 4909 (b) 4489 (c) 5329 (d) 5339
13. 13.If a + b = 15 and ab = 50, what is a² + b²? (a) 325 (b) 125 (c) 175 (d) 225
14. 14.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 15 (c) 16 (d) 56
15. 15.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 72 (c) 17 (d) 73
16. 16.If a + b = 21 and ab = 108, what is (a − b)²? (a) 3 (b) 441 (c) 225 (d) 9
17. 17.What is the sum of the roots of x² − 15x + 54 = 0? (a) 14 (b) 54 (c) 15 (d) 16
18. 18.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4901 (b) 5051 (c) 5041 (d) 4761
19. 19.What are the roots of x² − 9x + 18 = 0? (a) 2 and 9 (b) 3 and 6 (c) 1 and 8 (d) 1 and 18
20. 20.Which is the correct factorisation of x² + 10x + 24? (a) (x + 4)(x + 6) (b) (x + 1)(x + 9) (c) (x − 4)(x − 6) (d) (x + 1)(x + 24)
21. 21.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1609 (b) 1859 (c) 1369 (d) 1849
22. 22.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)
23. 23.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x − 6)(x − 9) (c) (x + 6)(x + 9) (d) (x + 1)(x + 54)
24. 24.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 17 (c) 72 (d) 16
25. 25.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2525 (b) 3025 (c) 3035 (d) 2025
26. 26.If a + b = 10 and ab = 24, what is (a − b)²? (a) 52 (b) 2 (c) 4 (d) 100
27. 27.If a + b = 12 and ab = 32, what is (a − b)²? (a) 16 (b) 144 (c) 4 (d) 80
28. 28.What are the roots of x² − 11x + 28 = 0? (a) 1 and 28 (b) 4 and 7 (c) 1 and 10 (d) 2 and 14
29. 29.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 18) (b) (x + 1)(x + 10) (c) (x − 2)(x − 9) (d) (x + 2)(x + 9)
30. 30.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1034 (b) 904 (c) 1024 (d) 784
31. 31.What is the sum of the roots of x² − 14x + 45 = 0? (a) 15 (b) 14 (c) 45 (d) 13
32. 32.If a + b = 9 and ab = 14, what is a² + b²? (a) 109 (b) 67 (c) 81 (d) 53
33. 33.If a + b = 14 and ab = 48, what is a² + b²? (a) 292 (b) 148 (c) 100 (d) 196
34. 34.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 7)(x + 8) (c) (x + 1)(x + 14) (d) (x + 1)(x + 56)
35. 35.If a + b = 7 and ab = 6, what is (a − b)²? (a) 5 (b) 49 (c) 37 (d) 25
36. 36.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6724 (b) 6404 (c) 6734 (d) 6084
37. 37.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 1)(x + 8) (c) (x + 1)(x + 5) (d) (x + 2)(x + 4)
38. 38.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x + 1)(x + 48) (c) (x − 6)(x − 8) (d) (x + 1)(x + 13)
39. 39.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3481 (b) 3731 (c) 3721 (d) 3601
40. 40.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 8125 (b) 9035 (c) 7225 (d) 9025

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