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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 857 · 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 = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 9 (c) 441 (d) 3
  2. 2.What are the roots of x² − 13x + 36 = 0? (a) 4 and 9 (b) 1 and 36 (c) 2 and 18 (d) 1 and 12
  3. 3.If a + b = 18 and ab = 72, what is a² + b²? (a) 468 (b) 324 (c) 180 (d) 252
  4. 4.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x + 3)(x + 6) (c) (x + 1)(x + 18) (d) (x − 3)(x − 6)
  5. 5.If a + b = 13 and ab = 30, what is a² + b²? (a) 169 (b) 109 (c) 139 (d) 229
  6. 6.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)
  7. 7.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x − 7)(x − 8) (c) (x + 1)(x + 14) (d) (x + 1)(x + 56)
  8. 8.If a + b = 18 and ab = 80, what is a² + b²? (a) 244 (b) 324 (c) 484 (d) 164
  9. 9.What is the product of the roots of x² − 13x + 42 = 0? (a) 13 (b) 42 (c) 41 (d) 43
  10. 10.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 271 (c) 1 (d) 361
  11. 11.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2809 (b) 2209 (c) 2819 (d) 2509
  12. 12.If a + b = 12 and ab = 20, what is (a − b)²? (a) 144 (b) 64 (c) 104 (d) 8
  13. 13.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 11035 (b) 10025 (c) 11025 (d) 9025
  14. 14.If a + b = 9 and ab = 18, what is (a − b)²? (a) 81 (b) 45 (c) 3 (d) 9
  15. 15.What is the sum of the roots of x² − 8x + 15 = 0? (a) 9 (b) 7 (c) 15 (d) 8
  16. 16.If a + b = 13 and ab = 36, what is a² + b²? (a) 133 (b) 169 (c) 97 (d) 241
  17. 17.If a + b = 17 and ab = 70, what is (a − b)²? (a) 3 (b) 149 (c) 289 (d) 9
  18. 18.If a + b = 10 and ab = 9, what is (a − b)²? (a) 82 (b) 64 (c) 100 (d) 8
  19. 19.What is the product of the roots of x² − 14x + 48 = 0? (a) 47 (b) 49 (c) 48 (d) 14
  20. 20.If a + b = 13 and ab = 12, what is (a − b)²? (a) 145 (b) 121 (c) 169 (d) 11
  21. 21.If a + b = 6 and ab = 5, what is (a − b)²? (a) 26 (b) 36 (c) 4 (d) 16
  22. 22.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x − 7)(x − 9) (c) (x + 7)(x + 9) (d) (x + 1)(x + 15)
  23. 23.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8836 (b) 7396 (c) 8116 (d) 8846
  24. 24.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 404 (b) 484 (c) 324 (d) 494
  25. 25.If a + b = 16 and ab = 63, what is (a − b)²? (a) 130 (b) 256 (c) 4 (d) 2
  26. 26.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 400 (c) 301 (d) 202
  27. 27.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1034 (b) 1024 (c) 904 (d) 784
  28. 28.If a + b = 16 and ab = 60, what is (a − b)²? (a) 4 (b) 256 (c) 16 (d) 136
  29. 29.What is the product of the roots of x² − 6x + 8 = 0? (a) 9 (b) 8 (c) 6 (d) 7
  30. 30.What are the roots of x² − 13x + 42 = 0? (a) 1 and 12 (b) 6 and 7 (c) 2 and 21 (d) 1 and 42
  31. 31.If a + b = 15 and ab = 36, what is (a − b)²? (a) 225 (b) 153 (c) 81 (d) 9
  32. 32.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)
  33. 33.What is the sum of the roots of x² − 12x + 32 = 0? (a) 12 (b) 11 (c) 13 (d) 32
  34. 34.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4366 (b) 5476 (c) 4884 (d) 4356
  35. 35.If a + b = 11 and ab = 10, what is a² + b²? (a) 101 (b) 141 (c) 121 (d) 111
  36. 36.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1521 (b) 1691 (c) 1601 (d) 1681
  37. 37.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1306 (b) 1584 (c) 1296 (d) 1936
  38. 38.What is the product of the roots of x² − 13x + 40 = 0? (a) 41 (b) 13 (c) 40 (d) 39
  39. 39.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1099 (b) 909 (c) 1089 (d) 729
  40. 40.What is the sum of the roots of x² − 11x + 24 = 0? (a) 11 (b) 10 (c) 12 (d) 24

Answer key

  1. 9
  2. 4 and 9
  3. 180
  4. (x + 3)(x + 6)
  5. 109
  6. (x + 6)(x + 7)
  7. (x + 7)(x + 8)
  8. 164
  9. 42
  10. 1
  11. 2809
  12. 64
  13. 11025
  14. 9
  15. 8
  16. 97
  17. 9
  18. 64
  19. 48
  20. 121
  21. 16
  22. (x + 7)(x + 9)
  23. 8836
  24. 484
  25. 4
  26. 202
  27. 1024
  28. 16
  29. 8
  30. 6 and 7
  31. 81
  32. (x + 8)(x + 9)
  33. 12
  34. 4356
  35. 101
  36. 1681
  37. 1296
  38. 40
  39. 1089
  40. 11

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