Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 661 · 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.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x − 6)(x − 9) (c) (x + 1)(x + 14) (d) (x + 1)(x + 54)
2. 2.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x + 3)(x + 9) (c) (x − 3)(x − 9) (d) (x + 1)(x + 27)
3. 3.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 5635 (b) 7225 (c) 5625 (d) 6375
4. 4.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 1024 (b) 784 (c) 794 (d) 896
5. 5.If a + b = 6 and ab = 5, what is a² + b²? (a) 31 (b) 26 (c) 36 (d) 46
6. 6.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3136 (b) 3616 (c) 4106 (d) 4096
7. 7.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3035 (b) 4225 (c) 3025 (d) 3575
8. 8.What is the product of the roots of x² − 11x + 18 = 0? (a) 11 (b) 18 (c) 17 (d) 19
9. 9.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 1)(x + 30) (c) (x + 1)(x + 10) (d) (x + 5)(x + 6)
10. 10.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 6)(x + 8) (c) (x − 6)(x − 8) (d) (x + 1)(x + 48)
11. 11.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
12. 12.If a + b = 20 and ab = 99, what is (a − b)²? (a) 4 (b) 400 (c) 202 (d) 2
13. 13.What is the product of the roots of x² − 13x + 40 = 0? (a) 13 (b) 39 (c) 41 (d) 40
14. 14.What are the roots of x² − 6x + 8 = 0? (a) 2 and 5 (b) 1 and 8 (c) 2 and 4 (d) 1 and 5
15. 15.What is the product of the roots of x² − 11x + 30 = 0? (a) 30 (b) 29 (c) 31 (d) 11
16. 16.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8281 (b) 8291 (c) 8101 (d) 7921
17. 17.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10826 (b) 9216 (c) 10816 (d) 10016
18. 18.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 1)(x + 14) (c) (x + 1)(x + 8) (d) (x + 2)(x + 7)
19. 19.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 361 (b) 441 (c) 451 (d) 401
20. 20.If a + b = 13 and ab = 42, what is (a − b)²? (a) 169 (b) 1 (c) 127 (d) 85
21. 21.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 1)(x + 36) (c) (x + 4)(x + 9) (d) (x + 1)(x + 12)
22. 22.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 56 (c) 15 (d) 57
23. 23.What is the sum of the roots of x² − 10x + 21 = 0? (a) 11 (b) 10 (c) 21 (d) 9
24. 24.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 16) (c) (x + 1)(x + 72) (d) (x − 8)(x − 9)
25. 25.If a + b = 12 and ab = 20, what is (a − b)²? (a) 144 (b) 8 (c) 64 (d) 104
26. 26.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 333 (c) 657 (d) 225
27. 27.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x − 2)(x − 3) (c) (x + 1)(x + 6) (d) (x + 2)(x + 3)
28. 28.What is the product of the roots of x² − 16x + 63 = 0? (a) 62 (b) 16 (c) 63 (d) 64
29. 29.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x − 6)(x − 7) (c) (x + 1)(x + 42) (d) (x + 1)(x + 12)
30. 30.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 12) (c) (x + 1)(x + 40) (d) (x + 5)(x + 8)
31. 31.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8109 (b) 7569 (c) 8649 (d) 8659
32. 32.What are the roots of x² − 15x + 54 = 0? (a) 6 and 9 (b) 1 and 14 (c) 1 and 54 (d) 2 and 27
33. 33.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 5329 (b) 4909 (c) 5339 (d) 4489
34. 34.If a + b = 10 and ab = 24, what is (a − b)²? (a) 52 (b) 4 (c) 2 (d) 100
35. 35.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 7)(x + 8) (c) (x − 7)(x − 8) (d) (x + 1)(x + 56)
36. 36.What is the sum of the roots of x² − 12x + 32 = 0? (a) 11 (b) 13 (c) 32 (d) 12
37. 37.If a + b = 12 and ab = 20, what is a² + b²? (a) 144 (b) 104 (c) 124 (d) 184
38. 38.What is the sum of the roots of x² − 13x + 42 = 0? (a) 13 (b) 42 (c) 14 (d) 12
39. 39.What is the sum of the roots of x² − 13x + 40 = 0? (a) 14 (b) 40 (c) 13 (d) 12
40. 40.If a + b = 13 and ab = 36, what is a² + b²? (a) 169 (b) 241 (c) 97 (d) 133

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