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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 396 · 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.If a + b = 14 and ab = 24, what is (a − b)²? (a) 196 (b) 10 (c) 148 (d) 100
  2. 2.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7066 (b) 7056 (c) 5776 (d) 6416
  3. 3.What is the product of the roots of x² − 9x + 18 = 0? (a) 17 (b) 18 (c) 9 (d) 19
  4. 4.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x + 1)(x + 8) (c) (x − 2)(x − 7) (d) (x + 2)(x + 7)
  5. 5.What is the product of the roots of x² − 6x + 8 = 0? (a) 9 (b) 8 (c) 6 (d) 7
  6. 6.If a + b = 11 and ab = 28, what is (a − b)²? (a) 65 (b) 3 (c) 121 (d) 9
  7. 7.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
  8. 8.If a + b = 19 and ab = 88, what is a² + b²? (a) 361 (b) 185 (c) 537 (d) 273
  9. 9.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1936 (b) 1306 (c) 1296 (d) 1584
  10. 10.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2809 (b) 2819 (c) 2509 (d) 2209
  11. 11.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 3)(x + 5) (c) (x + 1)(x + 15) (d) (x + 1)(x + 7)
  12. 12.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4904 (b) 5184 (c) 4624 (d) 5194
  13. 13.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6571 (b) 6561 (c) 6241 (d) 6401
  14. 14.What is the sum of the roots of x² − 7x + 12 = 0? (a) 7 (b) 12 (c) 8 (d) 6
  15. 15.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2916 (b) 2116 (c) 2126 (d) 2484
  16. 16.If a + b = 11 and ab = 30, what is a² + b²? (a) 181 (b) 91 (c) 61 (d) 121
  17. 17.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 925 (b) 1235 (c) 625 (d) 1225
  18. 18.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 72 (c) 17 (d) 71
  19. 19.If a + b = 17 and ab = 60, what is a² + b²? (a) 409 (b) 229 (c) 289 (d) 169
  20. 20.What is the product of the roots of x² − 10x + 21 = 0? (a) 10 (b) 20 (c) 22 (d) 21
  21. 21.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)
  22. 22.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 56 (c) 15 (d) 57
  23. 23.What are the roots of x² − 15x + 56 = 0? (a) 2 and 28 (b) 1 and 14 (c) 1 and 56 (d) 7 and 8
  24. 24.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 8 and 9 (c) 2 and 36 (d) 1 and 16
  25. 25.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1604 (b) 1764 (c) 1444 (d) 1774
  26. 26.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9984 (b) 10816 (c) 9216 (d) 9226
  27. 27.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 1)(x + 10) (c) (x − 5)(x − 6) (d) (x + 5)(x + 6)
  28. 28.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 10201 (b) 9999 (c) 9811 (d) 9801
  29. 29.If a + b = 16 and ab = 63, what is a² + b²? (a) 193 (b) 256 (c) 130 (d) 382
  30. 30.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 18 (c) 17 (d) 72
  31. 31.What are the roots of x² − 11x + 24 = 0? (a) 1 and 24 (b) 1 and 10 (c) 3 and 8 (d) 2 and 12
  32. 32.What are the roots of x² − 11x + 28 = 0? (a) 2 and 14 (b) 1 and 10 (c) 4 and 7 (d) 1 and 28
  33. 33.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9409 (b) 10609 (c) 9991 (d) 9419
  34. 34.What is the product of the roots of x² − 7x + 12 = 0? (a) 11 (b) 12 (c) 13 (d) 7
  35. 35.If a + b = 13 and ab = 30, what is (a − b)²? (a) 109 (b) 49 (c) 169 (d) 7
  36. 36.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 529 (b) 391 (c) 299 (d) 289
  37. 37.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 63 (c) 16 (d) 62
  38. 38.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 784 (c) 1024 (d) 896
  39. 39.If a + b = 18 and ab = 77, what is a² + b²? (a) 324 (b) 247 (c) 478 (d) 170
  40. 40.What are the roots of x² − 15x + 54 = 0? (a) 1 and 54 (b) 6 and 9 (c) 2 and 27 (d) 1 and 14

Answer key

  1. 100
  2. 7056
  3. 18
  4. (x + 2)(x + 7)
  5. 8
  6. 9
  7. (x + 8)(x + 9)
  8. 185
  9. 1296
  10. 2809
  11. (x + 3)(x + 5)
  12. 5184
  13. 6561
  14. 7
  15. 2116
  16. 61
  17. 1225
  18. 72
  19. 169
  20. 21
  21. (x + 2)(x + 9)
  22. 56
  23. 7 and 8
  24. 8 and 9
  25. 1764
  26. 9216
  27. (x + 5)(x + 6)
  28. 9801
  29. 130
  30. 17
  31. 3 and 8
  32. 4 and 7
  33. 9409
  34. 12
  35. 49
  36. 289
  37. 63
  38. 784
  39. 170
  40. 6 and 9

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