Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 89 · 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.If a + b = 15 and ab = 50, what is a² + b²? (a) 125 (b) 325 (c) 175 (d) 225
2. 2.If a + b = 18 and ab = 80, what is (a − b)²? (a) 4 (b) 324 (c) 2 (d) 164
3. 3.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8474 (b) 7744 (c) 8464 (d) 8104
4. 4.What is the product of the roots of x² − 9x + 20 = 0? (a) 19 (b) 20 (c) 9 (d) 21
5. 5.If a + b = 17 and ab = 70, what is (a − b)²? (a) 3 (b) 149 (c) 289 (d) 9
6. 6.What is the product of the roots of x² − 7x + 12 = 0? (a) 11 (b) 12 (c) 7 (d) 13
7. 7.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 11035 (b) 10025 (c) 11025 (d) 9025
8. 8.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x − 5)(x − 7) (c) (x + 1)(x + 35) (d) (x + 5)(x + 7)
9. 9.What are the roots of x² − 11x + 24 = 0? (a) 2 and 12 (b) 1 and 10 (c) 1 and 24 (d) 3 and 8
10. 10.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)
11. 11.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 399 (b) 371 (c) 441 (d) 361
12. 12.If a + b = 13 and ab = 40, what is a² + b²? (a) 249 (b) 89 (c) 129 (d) 169
13. 13.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 2 and 28 (c) 1 and 14 (d) 7 and 8
14. 14.If a + b = 11 and ab = 28, what is (a − b)²? (a) 65 (b) 9 (c) 3 (d) 121
15. 15.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)
16. 16.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)
17. 17.What are the roots of x² − 9x + 14 = 0? (a) 2 and 7 (b) 3 and 6 (c) 1 and 8 (d) 1 and 14
18. 18.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 9216 (b) 10016 (c) 10826 (d) 10816
19. 19.What is the sum of the roots of x² − 15x + 54 = 0? (a) 16 (b) 54 (c) 14 (d) 15
20. 20.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3599 (b) 3481 (c) 3721 (d) 3491
21. 21.If a + b = 11 and ab = 28, what is a² + b²? (a) 121 (b) 65 (c) 177 (d) 93
22. 22.What are the roots of x² − 7x + 10 = 0? (a) 3 and 4 (b) 1 and 6 (c) 2 and 5 (d) 1 and 10
23. 23.What is the product of the roots of x² − 15x + 54 = 0? (a) 15 (b) 54 (c) 53 (d) 55
24. 24.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6409 (b) 6899 (c) 6889 (d) 5929
25. 25.If a + b = 11 and ab = 10, what is (a − b)²? (a) 101 (b) 9 (c) 81 (d) 121
26. 26.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2025 (b) 1225 (c) 2035 (d) 1625
27. 27.If a + b = 19 and ab = 88, what is (a − b)²? (a) 185 (b) 9 (c) 361 (d) 3
28. 28.If a + b = 13 and ab = 42, what is a² + b²? (a) 169 (b) 85 (c) 253 (d) 127
29. 29.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 15) (c) (x + 1)(x + 63) (d) (x − 7)(x − 9)
30. 30.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1681 (b) 1691 (c) 1521 (d) 1601
31. 31.If a + b = 19 and ab = 84, what is a² + b²? (a) 361 (b) 193 (c) 529 (d) 277
32. 32.If a + b = 15 and ab = 56, what is a² + b²? (a) 113 (b) 225 (c) 169 (d) 337
33. 33.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 6)(x + 7) (c) (x − 6)(x − 7) (d) (x + 1)(x + 12)
34. 34.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6094 (b) 6396 (c) 6724 (d) 6084
35. 35.If a + b = 17 and ab = 72, what is a² + b²? (a) 289 (b) 217 (c) 145 (d) 433
36. 36.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 686 (b) 884 (c) 1156 (d) 676
37. 37.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 3025 (b) 2525 (c) 3035 (d) 2025
38. 38.If a + b = 4 and ab = 3, what is a² + b²? (a) 10 (b) 16 (c) 22 (d) 13
39. 39.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 1 and 16 (c) 1 and 72 (d) 8 and 9
40. 40.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 5625 (b) 7235 (c) 7225 (d) 6425

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