Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 288 · 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 = 10 and ab = 9, what is (a − b)²? (a) 100 (b) 64 (c) 82 (d) 8
2. 2.What is the product of the roots of x² − 11x + 18 = 0? (a) 17 (b) 18 (c) 11 (d) 19
3. 3.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5041 (b) 5051 (c) 4901 (d) 4761
4. 4.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1034 (b) 1024 (c) 904 (d) 784
5. 5.What are the roots of x² − 8x + 15 = 0? (a) 3 and 5 (b) 2 and 6 (c) 1 and 7 (d) 1 and 15
6. 6.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1444 (b) 1764 (c) 1774 (d) 1604
7. 7.What is the product of the roots of x² − 13x + 42 = 0? (a) 41 (b) 43 (c) 13 (d) 42
8. 8.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x − 3)(x − 7) (c) (x + 3)(x + 7) (d) (x + 1)(x + 9)
9. 9.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 2025 (b) 1225 (c) 1235 (d) 1575
10. 10.Which is the correct factorisation of x² + 8x + 12? (a) (x − 2)(x − 6) (b) (x + 2)(x + 6) (c) (x + 1)(x + 12) (d) (x + 1)(x + 7)
11. 11.If a + b = 13 and ab = 30, what is (a − b)²? (a) 49 (b) 109 (c) 169 (d) 7
12. 12.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 56) (c) (x + 7)(x + 8) (d) (x + 1)(x + 14)
13. 13.What is the sum of the roots of x² − 9x + 20 = 0? (a) 8 (b) 9 (c) 20 (d) 10
14. 14.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8091 (b) 7569 (c) 7579 (d) 8649
15. 15.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 2)(x + 4) (c) (x + 1)(x + 5) (d) (x − 2)(x − 4)
16. 16.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1946 (b) 1936 (c) 1616 (d) 1296
17. 17.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 576 (b) 586 (c) 256 (d) 416
18. 18.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x + 1)(x + 13) (c) (x + 6)(x + 8) (d) (x − 6)(x − 8)
19. 19.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x + 3)(x + 4) (c) (x − 3)(x − 4) (d) (x + 1)(x + 6)
20. 20.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x − 2)(x − 9) (c) (x + 2)(x + 9) (d) (x + 1)(x + 18)
21. 21.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 875 (b) 1225 (c) 635 (d) 625
22. 22.If a + b = 15 and ab = 56, what is (a − b)²? (a) 225 (b) 169 (c) 113 (d) 1
23. 23.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)
24. 24.If a + b = 14 and ab = 45, what is (a − b)²? (a) 196 (b) 106 (c) 4 (d) 16
25. 25.If a + b = 9 and ab = 14, what is (a − b)²? (a) 53 (b) 81 (c) 5 (d) 25
26. 26.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x + 1)(x + 54) (c) (x − 6)(x − 9) (d) (x + 6)(x + 9)
27. 27.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 1 and 72 (c) 8 and 9 (d) 1 and 16
28. 28.If a + b = 11 and ab = 24, what is a² + b²? (a) 169 (b) 73 (c) 97 (d) 121
29. 29.If a + b = 19 and ab = 88, what is (a − b)²? (a) 361 (b) 185 (c) 3 (d) 9
30. 30.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 625 (b) 375 (c) 225 (d) 235
31. 31.What are the roots of x² − 13x + 42 = 0? (a) 6 and 7 (b) 1 and 42 (c) 2 and 21 (d) 1 and 12
32. 32.What are the roots of x² − 15x + 54 = 0? (a) 2 and 27 (b) 6 and 9 (c) 1 and 54 (d) 1 and 14
33. 33.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 9025 (b) 11025 (c) 10025 (d) 11035
34. 34.What is the product of the roots of x² − 8x + 15 = 0? (a) 14 (b) 8 (c) 16 (d) 15
35. 35.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 529 (b) 289 (c) 391 (d) 299
36. 36.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2116 (b) 2126 (c) 2916 (d) 2484
37. 37.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x + 1)(x + 6) (c) (x − 2)(x − 5) (d) (x + 2)(x + 5)
38. 38.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 289 (c) 217 (d) 145
39. 39.If a + b = 15 and ab = 54, what is (a − b)²? (a) 117 (b) 3 (c) 225 (d) 9
40. 40.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3481 (b) 3721 (c) 3731 (d) 3601

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