Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 436 · 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.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8474 (b) 8104 (c) 8464 (d) 7744
2. 2.Which is the correct factorisation of x² + 11x + 18? (a) (x + 2)(x + 9) (b) (x + 1)(x + 18) (c) (x + 1)(x + 10) (d) (x − 2)(x − 9)
3. 3.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 484 (b) 324 (c) 334 (d) 396
4. 4.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)
5. 5.If a + b = 19 and ab = 84, what is a² + b²? (a) 529 (b) 277 (c) 361 (d) 193
6. 6.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9811 (b) 9999 (c) 10201 (d) 9801
7. 7.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4356 (b) 4916 (c) 5486 (d) 5476
8. 8.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x + 3)(x + 7) (c) (x + 1)(x + 9) (d) (x − 3)(x − 7)
9. 9.If a + b = 19 and ab = 88, what is a² + b²? (a) 361 (b) 537 (c) 273 (d) 185
10. 10.If a + b = 11 and ab = 28, what is (a − b)²? (a) 121 (b) 9 (c) 65 (d) 3
11. 11.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 7 and 8 (c) 1 and 14 (d) 2 and 28
12. 12.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 5329 (b) 4909 (c) 5339 (d) 4489
13. 13.If a + b = 9 and ab = 20, what is (a − b)²? (a) 41 (b) 81 (c) 1 (d) 61
14. 14.If a + b = 3 and ab = 2, what is a² + b²? (a) 9 (b) 13 (c) 7 (d) 5
15. 15.What is the sum of the roots of x² − 6x + 8 = 0? (a) 8 (b) 5 (c) 7 (d) 6
16. 16.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x + 1)(x + 40) (c) (x + 5)(x + 8) (d) (x − 5)(x − 8)
17. 17.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x + 1)(x + 12) (c) (x − 3)(x − 4) (d) (x + 3)(x + 4)
18. 18.If a + b = 7 and ab = 10, what is (a − b)²? (a) 49 (b) 9 (c) 29 (d) 3
19. 19.What is the sum of the roots of x² − 13x + 42 = 0? (a) 13 (b) 42 (c) 14 (d) 12
20. 20.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 1)(x + 6) (c) (x + 1)(x + 4) (d) (x + 2)(x + 3)
21. 21.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4884 (b) 5476 (c) 4366 (d) 4356
22. 22.What are the roots of x² − 16x + 63 = 0? (a) 1 and 15 (b) 7 and 9 (c) 3 and 21 (d) 1 and 63
23. 23.What is the sum of the roots of x² − 10x + 24 = 0? (a) 10 (b) 9 (c) 24 (d) 11
24. 24.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 529 (b) 289 (c) 539 (d) 409
25. 25.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 451 (b) 441 (c) 361 (d) 401
26. 26.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 56 (c) 15 (d) 57
27. 27.What is the product of the roots of x² − 12x + 35 = 0? (a) 36 (b) 35 (c) 12 (d) 34
28. 28.If a + b = 6 and ab = 5, what is a² + b²? (a) 26 (b) 31 (c) 36 (d) 46
29. 29.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 399 (b) 441 (c) 361 (d) 371
30. 30.What are the roots of x² − 13x + 40 = 0? (a) 2 and 20 (b) 1 and 12 (c) 1 and 40 (d) 5 and 8
31. 31.What is the sum of the roots of x² − 10x + 16 = 0? (a) 10 (b) 16 (c) 11 (d) 9
32. 32.If a + b = 13 and ab = 30, what is a² + b²? (a) 169 (b) 139 (c) 229 (d) 109
33. 33.If a + b = 8 and ab = 15, what is (a − b)²? (a) 34 (b) 64 (c) 2 (d) 4
34. 34.If a + b = 14 and ab = 40, what is (a − b)²? (a) 116 (b) 6 (c) 36 (d) 196
35. 35.What is the product of the roots of x² − 13x + 42 = 0? (a) 41 (b) 43 (c) 42 (d) 13
36. 36.What is the product of the roots of x² − 5x + 6 = 0? (a) 6 (b) 5 (c) 7 (d) 8
37. 37.If a + b = 4 and ab = 3, what is (a − b)²? (a) 10 (b) 16 (c) 4 (d) 2
38. 38.If a + b = 17 and ab = 70, what is (a − b)²? (a) 9 (b) 3 (c) 149 (d) 289
39. 39.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5041 (b) 5051 (c) 4761 (d) 4901
40. 40.If a + b = 18 and ab = 72, what is a² + b²? (a) 468 (b) 324 (c) 180 (d) 252

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