Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 678 · 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 − 10x + 21 = 0? (a) 3 and 7 (b) 1 and 21 (c) 2 and 8 (d) 1 and 9
2. 2.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 676 (b) 884 (c) 1156 (d) 686
3. 3.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 1)(x + 12) (c) (x + 6)(x + 7) (d) (x − 6)(x − 7)
4. 4.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 541 (c) 361 (d) 181
5. 5.What is the sum of the roots of x² − 9x + 20 = 0? (a) 10 (b) 8 (c) 9 (d) 20
6. 6.What is the sum of the roots of x² − 15x + 54 = 0? (a) 54 (b) 16 (c) 14 (d) 15
7. 7.What is the sum of the roots of x² − 6x + 8 = 0? (a) 6 (b) 7 (c) 8 (d) 5
8. 8.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 484 (b) 396 (c) 334 (d) 324
9. 9.If a + b = 18 and ab = 80, what is a² + b²? (a) 164 (b) 244 (c) 484 (d) 324
10. 10.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 8 and 9 (c) 1 and 72 (d) 2 and 36
11. 11.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 72) (c) (x − 8)(x − 9) (d) (x + 1)(x + 16)
12. 12.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 5625 (b) 6375 (c) 7225 (d) 5635
13. 13.If a + b = 17 and ab = 66, what is a² + b²? (a) 421 (b) 157 (c) 223 (d) 289
14. 14.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 1)(x + 10) (c) (x + 5)(x + 6) (d) (x + 1)(x + 30)
15. 15.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2809 (b) 2219 (c) 2209 (d) 2491
16. 16.If a + b = 13 and ab = 40, what is a² + b²? (a) 249 (b) 129 (c) 89 (d) 169
17. 17.What is the sum of the roots of x² − 11x + 24 = 0? (a) 10 (b) 24 (c) 11 (d) 12
18. 18.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 1)(x + 56) (c) (x + 7)(x + 8) (d) (x − 7)(x − 8)
19. 19.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 6)(x + 9) (c) (x + 1)(x + 54) (d) (x + 1)(x + 14)
20. 20.What are the roots of x² − 15x + 54 = 0? (a) 1 and 14 (b) 2 and 27 (c) 1 and 54 (d) 6 and 9
21. 21.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 14 (c) 56 (d) 15
22. 22.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x + 5)(x + 9) (c) (x + 1)(x + 45) (d) (x − 5)(x − 9)
23. 23.What is the product of the roots of x² − 5x + 6 = 0? (a) 5 (b) 6 (c) 7 (d) 8
24. 24.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9409 (b) 9991 (c) 9419 (d) 10609
25. 25.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6734 (b) 6084 (c) 6404 (d) 6724
26. 26.If a + b = 11 and ab = 10, what is a² + b²? (a) 141 (b) 121 (c) 101 (d) 111
27. 27.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 361 (b) 441 (c) 399 (d) 371
28. 28.If a + b = 18 and ab = 72, what is (a − b)²? (a) 36 (b) 6 (c) 180 (d) 324
29. 29.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6251 (b) 6241 (c) 6561 (d) 6399
30. 30.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 5476 (b) 4356 (c) 5486 (d) 4916
31. 31.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 271 (c) 361 (d) 1
32. 32.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2035 (b) 2025 (c) 1225 (d) 1625
33. 33.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 1)(x + 11) (c) (x + 5)(x + 7) (d) (x + 1)(x + 35)
34. 34.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 729 (b) 1089 (c) 909 (d) 1099
35. 35.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 27) (b) (x + 3)(x + 9) (c) (x − 3)(x − 9) (d) (x + 1)(x + 11)
36. 36.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2116 (b) 2916 (c) 2484 (d) 2126
37. 37.If a + b = 14 and ab = 40, what is a² + b²? (a) 156 (b) 116 (c) 276 (d) 196
38. 38.What are the roots of x² − 14x + 45 = 0? (a) 1 and 45 (b) 3 and 15 (c) 5 and 9 (d) 1 and 13
39. 39.What are the roots of x² − 9x + 20 = 0? (a) 1 and 20 (b) 4 and 5 (c) 1 and 8 (d) 2 and 10
40. 40.If a + b = 7 and ab = 10, what is (a − b)²? (a) 9 (b) 29 (c) 3 (d) 49

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