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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 563 · 40 questions
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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 15²? (a) 625 (b) 375 (c) 225 (d) 235
  2. 2.If a + b = 15 and ab = 56, what is (a − b)²? (a) 169 (b) 1 (c) 225 (d) 113
  3. 3.What is the sum of the roots of x² − 15x + 54 = 0? (a) 54 (b) 16 (c) 14 (d) 15
  4. 4.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1691 (b) 1521 (c) 1601 (d) 1681
  5. 5.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1156 (b) 676 (c) 916 (d) 1166
  6. 6.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
  7. 7.Which is the correct factorisation of x² + 12x + 27? (a) (x + 3)(x + 9) (b) (x + 1)(x + 11) (c) (x − 3)(x − 9) (d) (x + 1)(x + 27)
  8. 8.If a + b = 11 and ab = 28, what is a² + b²? (a) 121 (b) 65 (c) 177 (d) 93
  9. 9.If a + b = 12 and ab = 27, what is a² + b²? (a) 198 (b) 90 (c) 144 (d) 117
  10. 10.If a + b = 14 and ab = 45, what is a² + b²? (a) 286 (b) 151 (c) 196 (d) 106
  11. 11.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1444 (b) 1604 (c) 1774 (d) 1764
  12. 12.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
  13. 13.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1936 (b) 1296 (c) 1584 (d) 1306
  14. 14.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x − 5)(x − 6) (c) (x + 1)(x + 10) (d) (x + 5)(x + 6)
  15. 15.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4899 (b) 4771 (c) 4761 (d) 5041
  16. 16.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1454 (b) 1764 (c) 1444 (d) 1596
  17. 17.What are the roots of x² − 16x + 63 = 0? (a) 1 and 63 (b) 7 and 9 (c) 3 and 21 (d) 1 and 15
  18. 18.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4225 (b) 4235 (c) 5625 (d) 4875
  19. 19.If a + b = 4 and ab = 3, what is a² + b²? (a) 22 (b) 16 (c) 13 (d) 10
  20. 20.If a + b = 14 and ab = 48, what is (a − b)²? (a) 2 (b) 4 (c) 196 (d) 100
  21. 21.What are the roots of x² − 15x + 56 = 0? (a) 2 and 28 (b) 7 and 8 (c) 1 and 56 (d) 1 and 14
  22. 22.What is the product of the roots of x² − 10x + 16 = 0? (a) 16 (b) 10 (c) 17 (d) 15
  23. 23.If a + b = 17 and ab = 66, what is a² + b²? (a) 289 (b) 223 (c) 421 (d) 157
  24. 24.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 15 (c) 17 (d) 63
  25. 25.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 202 (c) 4 (d) 2
  26. 26.What is the product of the roots of x² − 13x + 40 = 0? (a) 39 (b) 41 (c) 13 (d) 40
  27. 27.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7056 (b) 6416 (c) 5776 (d) 7066
  28. 28.If a + b = 13 and ab = 42, what is (a − b)²? (a) 85 (b) 127 (c) 1 (d) 169
  29. 29.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4896 (b) 4634 (c) 5184 (d) 4624
  30. 30.Which is the correct factorisation of x² + 11x + 18? (a) (x − 2)(x − 9) (b) (x + 1)(x + 18) (c) (x + 2)(x + 9) (d) (x + 1)(x + 10)
  31. 31.If a + b = 15 and ab = 50, what is (a − b)²? (a) 5 (b) 225 (c) 25 (d) 125
  32. 32.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x − 5)(x − 9) (c) (x + 1)(x + 13) (d) (x + 5)(x + 9)
  33. 33.If a + b = 17 and ab = 72, what is a² + b²? (a) 289 (b) 217 (c) 433 (d) 145
  34. 34.If a + b = 17 and ab = 72, what is (a − b)²? (a) 289 (b) 145 (c) 1 (d) 217
  35. 35.What is the product of the roots of x² − 9x + 20 = 0? (a) 9 (b) 21 (c) 20 (d) 19
  36. 36.If a + b = 17 and ab = 60, what is a² + b²? (a) 409 (b) 169 (c) 289 (d) 229
  37. 37.If a + b = 17 and ab = 66, what is (a − b)²? (a) 289 (b) 157 (c) 5 (d) 25
  38. 38.If a + b = 13 and ab = 40, what is a² + b²? (a) 249 (b) 169 (c) 129 (d) 89
  39. 39.If a + b = 15 and ab = 54, what is (a − b)²? (a) 3 (b) 225 (c) 117 (d) 9
  40. 40.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 494 (b) 404 (c) 484 (d) 324

Answer key

  1. 225
  2. 1
  3. 15
  4. 1681
  5. 1156
  6. 8 and 9
  7. (x + 3)(x + 9)
  8. 65
  9. 90
  10. 106
  11. 1764
  12. (x + 8)(x + 9)
  13. 1296
  14. (x + 5)(x + 6)
  15. 4761
  16. 1444
  17. 7 and 9
  18. 4225
  19. 10
  20. 4
  21. 7 and 8
  22. 16
  23. 157
  24. 16
  25. 4
  26. 40
  27. 7056
  28. 1
  29. 4624
  30. (x + 2)(x + 9)
  31. 25
  32. (x + 5)(x + 9)
  33. 145
  34. 1
  35. 20
  36. 169
  37. 25
  38. 89
  39. 9
  40. 484

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