Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 907 · 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 product of the roots of x² − 13x + 36 = 0? (a) 37 (b) 13 (c) 35 (d) 36
2. 2.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
3. 3.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 541 (c) 181 (d) 361
4. 4.What is the sum of the roots of x² − 6x + 8 = 0? (a) 6 (b) 7 (c) 5 (d) 8
5. 5.If a + b = 17 and ab = 60, what is a² + b²? (a) 409 (b) 229 (c) 289 (d) 169
6. 6.If a + b = 10 and ab = 21, what is a² + b²? (a) 79 (b) 100 (c) 142 (d) 58
7. 7.If a + b = 9 and ab = 14, what is a² + b²? (a) 81 (b) 53 (c) 109 (d) 67
8. 8.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 8836 (b) 7406 (c) 7396 (d) 8084
9. 9.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x − 2)(x − 8) (c) (x + 2)(x + 8) (d) (x + 1)(x + 16)
10. 10.If a + b = 12 and ab = 35, what is (a − b)²? (a) 2 (b) 74 (c) 144 (d) 4
11. 11.If a + b = 4 and ab = 3, what is a² + b²? (a) 10 (b) 13 (c) 22 (d) 16
12. 12.If a + b = 4 and ab = 3, what is (a − b)²? (a) 16 (b) 2 (c) 4 (d) 10
13. 13.What is the product of the roots of x² − 13x + 42 = 0? (a) 41 (b) 42 (c) 13 (d) 43
14. 14.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 625 (b) 1225 (c) 635 (d) 875
15. 15.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x − 7)(x − 9) (c) (x + 1)(x + 63) (d) (x + 1)(x + 15)
16. 16.If a + b = 17 and ab = 72, what is (a − b)²? (a) 1 (b) 217 (c) 289 (d) 145
17. 17.Which is the correct factorisation of x² + 11x + 24? (a) (x + 3)(x + 8) (b) (x − 3)(x − 8) (c) (x + 1)(x + 10) (d) (x + 1)(x + 24)
18. 18.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x + 1)(x + 10) (c) (x + 4)(x + 7) (d) (x − 4)(x − 7)
19. 19.If a + b = 7 and ab = 10, what is a² + b²? (a) 29 (b) 39 (c) 69 (d) 49
20. 20.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8104 (b) 7744 (c) 8464 (d) 8474
21. 21.If a + b = 18 and ab = 72, what is a² + b²? (a) 468 (b) 252 (c) 180 (d) 324
22. 22.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6889 (b) 6899 (c) 5929 (d) 6409
23. 23.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4366 (b) 4356 (c) 5476 (d) 4884
24. 24.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 14 (c) 56 (d) 15
25. 25.Which is the correct factorisation of x² + 11x + 18? (a) (x + 2)(x + 9) (b) (x + 1)(x + 18) (c) (x − 2)(x − 9) (d) (x + 1)(x + 10)
26. 26.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1691 (b) 1681 (c) 1521 (d) 1601
27. 27.What are the roots of x² − 9x + 20 = 0? (a) 1 and 20 (b) 2 and 10 (c) 4 and 5 (d) 1 and 8
28. 28.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 54) (c) (x + 6)(x + 9) (d) (x + 1)(x + 14)
29. 29.If a + b = 20 and ab = 99, what is (a − b)²? (a) 202 (b) 400 (c) 4 (d) 2
30. 30.What are the roots of x² − 10x + 24 = 0? (a) 1 and 24 (b) 2 and 12 (c) 4 and 6 (d) 1 and 9
31. 31.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x − 5)(x − 6) (c) (x + 5)(x + 6) (d) (x + 1)(x + 30)
32. 32.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 3)(x + 4) (c) (x + 1)(x + 12) (d) (x + 1)(x + 6)
33. 33.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x + 5)(x + 8) (c) (x + 1)(x + 40) (d) (x − 5)(x − 8)
34. 34.What are the roots of x² − 12x + 27 = 0? (a) 2 and 10 (b) 3 and 9 (c) 1 and 27 (d) 1 and 11
35. 35.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 225 (b) 635 (c) 625 (d) 425
36. 36.What is the sum of the roots of x² − 14x + 48 = 0? (a) 13 (b) 14 (c) 15 (d) 48
37. 37.If a + b = 7 and ab = 6, what is a² + b²? (a) 37 (b) 49 (c) 43 (d) 61
38. 38.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 56 (c) 15 (d) 55
39. 39.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 916 (b) 676 (c) 1156 (d) 1166
40. 40.What are the roots of x² − 7x + 12 = 0? (a) 2 and 6 (b) 1 and 6 (c) 3 and 4 (d) 1 and 12

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