← Identities, polynomials and quadratic roots lesson New set →

Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 14 · 40 questions
TalentJR

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.

NameDateTime takenScore ___ / 40
  1. 1.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8649 (b) 7579 (c) 7569 (d) 8091
  2. 2.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1609 (b) 1859 (c) 1369 (d) 1849
  3. 3.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3591 (b) 3249 (c) 3969 (d) 3259
  4. 4.If a + b = 17 and ab = 70, what is a² + b²? (a) 289 (b) 149 (c) 219 (d) 429
  5. 5.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x − 3)(x − 6) (c) (x + 1)(x + 8) (d) (x + 3)(x + 6)
  6. 6.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 1)(x + 14) (c) (x − 6)(x − 9) (d) (x + 6)(x + 9)
  7. 7.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 7)(x + 9) (c) (x + 1)(x + 63) (d) (x − 7)(x − 9)
  8. 8.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9226 (b) 10816 (c) 9216 (d) 9984
  9. 9.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 266 (b) 384 (c) 256 (d) 576
  10. 10.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 3035 (b) 2025 (c) 3025 (d) 2525
  11. 11.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 4)(x + 6) (c) (x + 1)(x + 9) (d) (x + 1)(x + 24)
  12. 12.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 1)(x + 72) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
  13. 13.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3625 (b) 3025 (c) 4235 (d) 4225
  14. 14.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2499 (b) 2401 (c) 2411 (d) 2601
  15. 15.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 2025 (b) 1225 (c) 1235 (d) 1575
  16. 16.If a + b = 15 and ab = 56, what is a² + b²? (a) 337 (b) 225 (c) 113 (d) 169
  17. 17.If a + b = 21 and ab = 108, what is a² + b²? (a) 657 (b) 333 (c) 225 (d) 441
  18. 18.What is the product of the roots of x² − 13x + 40 = 0? (a) 41 (b) 13 (c) 39 (d) 40
  19. 19.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 16 (c) 15 (d) 56
  20. 20.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 1225 (b) 635 (c) 625 (d) 875
  21. 21.If a + b = 14 and ab = 24, what is a² + b²? (a) 172 (b) 244 (c) 148 (d) 196
  22. 22.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x + 2)(x + 6) (c) (x − 2)(x − 6) (d) (x + 1)(x + 7)
  23. 23.What are the roots of x² − 10x + 21 = 0? (a) 2 and 8 (b) 1 and 21 (c) 3 and 7 (d) 1 and 9
  24. 24.If a + b = 11 and ab = 24, what is a² + b²? (a) 73 (b) 121 (c) 97 (d) 169
  25. 25.If a + b = 16 and ab = 60, what is a² + b²? (a) 376 (b) 136 (c) 196 (d) 256
  26. 26.If a + b = 13 and ab = 30, what is a² + b²? (a) 109 (b) 229 (c) 139 (d) 169
  27. 27.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 541 (c) 181 (d) 271
  28. 28.If a + b = 15 and ab = 50, what is (a − b)²? (a) 5 (b) 125 (c) 25 (d) 225
  29. 29.If a + b = 11 and ab = 30, what is (a − b)²? (a) 61 (b) 91 (c) 1 (d) 121
  30. 30.If a + b = 12 and ab = 27, what is (a − b)²? (a) 90 (b) 144 (c) 6 (d) 36
  31. 31.If a + b = 20 and ab = 99, what is a² + b²? (a) 400 (b) 301 (c) 202 (d) 598
  32. 32.If a + b = 12 and ab = 27, what is a² + b²? (a) 144 (b) 90 (c) 198 (d) 117
  33. 33.If a + b = 10 and ab = 16, what is (a − b)²? (a) 68 (b) 36 (c) 100 (d) 6
  34. 34.What is the sum of the roots of x² − 15x + 54 = 0? (a) 15 (b) 54 (c) 16 (d) 14
  35. 35.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 18 (c) 16 (d) 72
  36. 36.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 5476 (b) 4366 (c) 4884 (d) 4356
  37. 37.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 484 (b) 324 (c) 494 (d) 404
  38. 38.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4634 (b) 4624 (c) 4896 (d) 5184
  39. 39.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3136 (b) 3146 (c) 4096 (d) 3584
  40. 40.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10004 (b) 10404 (c) 10414 (d) 9604

Answer key

  1. 7569
  2. 1849
  3. 3249
  4. 149
  5. (x + 3)(x + 6)
  6. (x + 6)(x + 9)
  7. (x + 7)(x + 9)
  8. 9216
  9. 256
  10. 3025
  11. (x + 4)(x + 6)
  12. (x + 8)(x + 9)
  13. 4225
  14. 2401
  15. 1225
  16. 113
  17. 225
  18. 40
  19. 15
  20. 625
  21. 148
  22. (x + 2)(x + 6)
  23. 3 and 7
  24. 73
  25. 136
  26. 109
  27. 181
  28. 25
  29. 1
  30. 36
  31. 202
  32. 90
  33. 36
  34. 15
  35. 17
  36. 4356
  37. 484
  38. 4624
  39. 3136
  40. 10404

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