Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 827 · 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.What is the sum of the roots of x² − 12x + 27 = 0? (a) 13 (b) 11 (c) 12 (d) 27
2. 2.What is the sum of the roots of x² − 13x + 36 = 0? (a) 36 (b) 14 (c) 12 (d) 13
3. 3.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 15 (c) 16 (d) 17
4. 4.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9409 (b) 9991 (c) 10609 (d) 9419
5. 5.If a + b = 12 and ab = 32, what is (a − b)²? (a) 4 (b) 80 (c) 144 (d) 16
6. 6.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)
7. 7.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 1)(x + 12) (c) (x + 6)(x + 7) (d) (x − 6)(x − 7)
8. 8.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10016 (b) 10826 (c) 10816 (d) 9216
9. 9.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 529 (b) 409 (c) 539 (d) 289
10. 10.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 7)(x + 9) (c) (x + 1)(x + 15) (d) (x + 1)(x + 63)
11. 11.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 289 (b) 299 (c) 529 (d) 391
12. 12.If a + b = 16 and ab = 55, what is (a − b)²? (a) 256 (b) 146 (c) 36 (d) 6
13. 13.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 1089 (b) 729 (c) 739 (d) 891
14. 14.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 1225 (b) 875 (c) 625 (d) 635
15. 15.If a + b = 3 and ab = 2, what is (a − b)²? (a) 5 (b) 1 (c) 7 (d) 9
16. 16.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1369 (b) 1609 (c) 1849 (d) 1859
17. 17.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 72 (c) 17 (d) 71
18. 18.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 17 (c) 18 (d) 72
19. 19.What are the roots of x² − 12x + 35 = 0? (a) 5 and 7 (b) 1 and 11 (c) 2 and 10 (d) 1 and 35
20. 20.If a + b = 15 and ab = 44, what is (a − b)²? (a) 225 (b) 137 (c) 49 (d) 7
21. 21.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)
22. 22.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 8836 (b) 8084 (c) 7396 (d) 7406
23. 23.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)
24. 24.If a + b = 18 and ab = 77, what is a² + b²? (a) 247 (b) 478 (c) 324 (d) 170
25. 25.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x + 1)(x + 27) (c) (x − 3)(x − 9) (d) (x + 3)(x + 9)
26. 26.What are the roots of x² − 6x + 8 = 0? (a) 2 and 5 (b) 1 and 5 (c) 2 and 4 (d) 1 and 8
27. 27.Which is the correct factorisation of x² + 10x + 24? (a) (x + 4)(x + 6) (b) (x − 4)(x − 6) (c) (x + 1)(x + 9) (d) (x + 1)(x + 24)
28. 28.If a + b = 7 and ab = 12, what is a² + b²? (a) 73 (b) 49 (c) 37 (d) 25
29. 29.If a + b = 11 and ab = 30, what is (a − b)²? (a) 61 (b) 91 (c) 121 (d) 1
30. 30.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 11) (c) (x + 4)(x + 8) (d) (x + 1)(x + 32)
31. 31.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 225 (b) 625 (c) 635 (d) 425
32. 32.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3721 (b) 3731 (c) 3601 (d) 3481
33. 33.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)
34. 34.If a + b = 20 and ab = 96, what is (a − b)²? (a) 208 (b) 4 (c) 16 (d) 400
35. 35.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3854 (b) 3364 (c) 3604 (d) 3844
36. 36.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x − 3)(x − 6) (c) (x + 1)(x + 18) (d) (x + 3)(x + 6)
37. 37.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 1 and 72 (c) 2 and 36 (d) 8 and 9
38. 38.If a + b = 5 and ab = 4, what is (a − b)²? (a) 3 (b) 9 (c) 25 (d) 17
39. 39.What are the roots of x² − 13x + 42 = 0? (a) 1 and 12 (b) 2 and 21 (c) 1 and 42 (d) 6 and 7
40. 40.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3616 (b) 3136 (c) 4106 (d) 4096

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