Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 605 · 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 are the roots of x² − 13x + 36 = 0? (a) 2 and 18 (b) 4 and 9 (c) 1 and 12 (d) 1 and 36
2. 2.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10826 (b) 10816 (c) 10016 (d) 9216
3. 3.What is the sum of the roots of x² − 9x + 14 = 0? (a) 9 (b) 8 (c) 14 (d) 10
4. 4.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 73 (c) 17 (d) 72
5. 5.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 14) (c) (x + 1)(x + 56) (d) (x + 7)(x + 8)
6. 6.If a + b = 15 and ab = 36, what is (a − b)²? (a) 225 (b) 81 (c) 9 (d) 153
7. 7.What is the product of the roots of x² − 9x + 20 = 0? (a) 21 (b) 20 (c) 9 (d) 19
8. 8.What is the product of the roots of x² − 5x + 6 = 0? (a) 5 (b) 6 (c) 7 (d) 8
9. 9.If a + b = 14 and ab = 45, what is a² + b²? (a) 151 (b) 196 (c) 286 (d) 106
10. 10.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 401 (b) 361 (c) 451 (d) 441
11. 11.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 841 (b) 851 (c) 899 (d) 961
12. 12.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x − 5)(x − 7) (c) (x + 5)(x + 7) (d) (x + 1)(x + 11)
13. 13.What is the product of the roots of x² − 10x + 21 = 0? (a) 10 (b) 21 (c) 20 (d) 22
14. 14.What is the product of the roots of x² − 13x + 36 = 0? (a) 37 (b) 35 (c) 36 (d) 13
15. 15.If a + b = 12 and ab = 32, what is a² + b²? (a) 144 (b) 80 (c) 208 (d) 112
16. 16.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 4106 (b) 4096 (c) 3136 (d) 3616
17. 17.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x − 6)(x − 8) (c) (x + 1)(x + 48) (d) (x + 1)(x + 13)
18. 18.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2116 (b) 2516 (c) 2926 (d) 2916
19. 19.What is the product of the roots of x² − 14x + 48 = 0? (a) 48 (b) 14 (c) 49 (d) 47
20. 20.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2501 (b) 2611 (c) 2401 (d) 2601
21. 21.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x + 1)(x + 16)
22. 22.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9984 (b) 9226 (c) 10816 (d) 9216
23. 23.What is the product of the roots of x² − 7x + 10 = 0? (a) 11 (b) 7 (c) 9 (d) 10
24. 24.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7754 (b) 7744 (c) 8096 (d) 8464
25. 25.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x − 2)(x − 7) (c) (x + 1)(x + 8) (d) (x + 2)(x + 7)
26. 26.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 400 (c) 202 (d) 301
27. 27.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 27) (b) (x − 3)(x − 9) (c) (x + 1)(x + 11) (d) (x + 3)(x + 9)
28. 28.If a + b = 8 and ab = 12, what is a² + b²? (a) 64 (b) 40 (c) 52 (d) 88
29. 29.If a + b = 11 and ab = 28, what is (a − b)²? (a) 9 (b) 3 (c) 121 (d) 65
30. 30.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 625 (b) 235 (c) 375 (d) 225
31. 31.What are the roots of x² − 10x + 21 = 0? (a) 1 and 9 (b) 3 and 7 (c) 1 and 21 (d) 2 and 8
32. 32.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 72 (c) 18 (d) 17
33. 33.What is the product of the roots of x² − 6x + 8 = 0? (a) 8 (b) 9 (c) 7 (d) 6
34. 34.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7235 (b) 9025 (c) 7225 (d) 8075
35. 35.If a + b = 17 and ab = 66, what is (a − b)²? (a) 157 (b) 289 (c) 5 (d) 25
36. 36.If a + b = 12 and ab = 35, what is a² + b²? (a) 74 (b) 214 (c) 109 (d) 144
37. 37.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 1 and 16 (c) 8 and 9 (d) 1 and 72
38. 38.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x + 1)(x + 8) (c) (x + 3)(x + 6) (d) (x − 3)(x − 6)
39. 39.What is the product of the roots of x² − 11x + 30 = 0? (a) 31 (b) 30 (c) 11 (d) 29
40. 40.If a + b = 8 and ab = 15, what is (a − b)²? (a) 2 (b) 4 (c) 34 (d) 64

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