Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 131 · 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) 1 and 36 (b) 1 and 12 (c) 4 and 9 (d) 2 and 18
2. 2.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)
3. 3.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9996 (b) 10404 (c) 9614 (d) 9604
4. 4.If a + b = 8 and ab = 15, what is (a − b)²? (a) 64 (b) 2 (c) 34 (d) 4
5. 5.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x − 3)(x − 7) (c) (x + 3)(x + 7) (d) (x + 1)(x + 21)
6. 6.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 1156 (b) 676 (c) 686 (d) 884
7. 7.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 441 (b) 399 (c) 361 (d) 371
8. 8.What are the roots of x² − 11x + 28 = 0? (a) 1 and 28 (b) 4 and 7 (c) 2 and 14 (d) 1 and 10
9. 9.What are the roots of x² − 12x + 27 = 0? (a) 2 and 10 (b) 1 and 27 (c) 1 and 11 (d) 3 and 9
10. 10.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)
11. 11.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 5476 (b) 4356 (c) 4884 (d) 4366
12. 12.What are the roots of x² − 10x + 21 = 0? (a) 3 and 7 (b) 1 and 21 (c) 1 and 9 (d) 2 and 8
13. 13.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x + 1)(x + 13) (c) (x − 5)(x − 9) (d) (x + 5)(x + 9)
14. 14.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10009 (b) 10609 (c) 9409 (d) 10619
15. 15.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x + 1)(x + 35) (c) (x − 5)(x − 7) (d) (x + 5)(x + 7)
16. 16.What is the product of the roots of x² − 10x + 21 = 0? (a) 10 (b) 22 (c) 21 (d) 20
17. 17.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x + 1)(x + 8) (c) (x + 4)(x + 5) (d) (x − 4)(x − 5)
18. 18.What is the product of the roots of x² − 14x + 45 = 0? (a) 14 (b) 45 (c) 44 (d) 46
19. 19.Which is the correct factorisation of x² + 11x + 18? (a) (x + 2)(x + 9) (b) (x + 1)(x + 10) (c) (x + 1)(x + 18) (d) (x − 2)(x − 9)
20. 20.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2219 (b) 2809 (c) 2491 (d) 2209
21. 21.If a + b = 15 and ab = 36, what is (a − b)²? (a) 9 (b) 81 (c) 225 (d) 153
22. 22.If a + b = 13 and ab = 42, what is (a − b)²? (a) 85 (b) 127 (c) 169 (d) 1
23. 23.If a + b = 14 and ab = 33, what is (a − b)²? (a) 130 (b) 196 (c) 8 (d) 64
24. 24.If a + b = 18 and ab = 77, what is (a − b)²? (a) 324 (b) 4 (c) 16 (d) 170
25. 25.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1379 (b) 1591 (c) 1849 (d) 1369
26. 26.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8109 (b) 8659 (c) 8649 (d) 7569
27. 27.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8291 (b) 8281 (c) 8101 (d) 7921
28. 28.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 6)(x + 9) (c) (x − 6)(x − 9) (d) (x + 1)(x + 14)
29. 29.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 5929 (b) 6889 (c) 5939 (d) 6391
30. 30.What is the product of the roots of x² − 16x + 63 = 0? (a) 62 (b) 64 (c) 16 (d) 63
31. 31.If a + b = 19 and ab = 84, what is (a − b)²? (a) 193 (b) 25 (c) 361 (d) 5
32. 32.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7754 (b) 8096 (c) 8464 (d) 7744
33. 33.If a + b = 16 and ab = 63, what is (a − b)²? (a) 256 (b) 130 (c) 4 (d) 2
34. 34.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4901 (b) 5051 (c) 4761 (d) 5041
35. 35.What is the product of the roots of x² − 11x + 28 = 0? (a) 29 (b) 28 (c) 27 (d) 11
36. 36.If a + b = 19 and ab = 90, what is (a − b)²? (a) 1 (b) 181 (c) 271 (d) 361
37. 37.What are the roots of x² − 11x + 24 = 0? (a) 2 and 12 (b) 1 and 10 (c) 3 and 8 (d) 1 and 24
38. 38.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9991 (b) 10609 (c) 9419 (d) 9409
39. 39.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 7569 (b) 8091 (c) 8649 (d) 7579
40. 40.If a + b = 18 and ab = 80, what is (a − b)²? (a) 4 (b) 2 (c) 164 (d) 324

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