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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 797 · 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.What are the roots of x² − 13x + 42 = 0? (a) 1 and 12 (b) 2 and 21 (c) 6 and 7 (d) 1 and 42
  2. 2.If a + b = 19 and ab = 88, what is a² + b²? (a) 185 (b) 361 (c) 537 (d) 273
  3. 3.If a + b = 8 and ab = 7, what is (a − b)²? (a) 64 (b) 6 (c) 36 (d) 50
  4. 4.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7225 (b) 5625 (c) 6425 (d) 7235
  5. 5.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
  6. 6.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1599 (b) 1531 (c) 1681 (d) 1521
  7. 7.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 441 (c) 9 (d) 3
  8. 8.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2819 (b) 2809 (c) 2509 (d) 2209
  9. 9.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8464 (b) 8104 (c) 8474 (d) 7744
  10. 10.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3604 (b) 3364 (c) 3844 (d) 3854
  11. 11.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 7921 (b) 8101 (c) 8291 (d) 8281
  12. 12.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 56 (c) 57 (d) 15
  13. 13.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x + 1)(x + 13) (c) (x + 1)(x + 45) (d) (x − 5)(x − 9)
  14. 14.If a + b = 18 and ab = 77, what is (a − b)²? (a) 324 (b) 4 (c) 16 (d) 170
  15. 15.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 539 (b) 409 (c) 529 (d) 289
  16. 16.If a + b = 14 and ab = 48, what is a² + b²? (a) 148 (b) 100 (c) 292 (d) 196
  17. 17.If a + b = 17 and ab = 66, what is (a − b)²? (a) 289 (b) 5 (c) 157 (d) 25
  18. 18.If a + b = 5 and ab = 4, what is a² + b²? (a) 21 (b) 33 (c) 17 (d) 25
  19. 19.If a + b = 5 and ab = 6, what is a² + b²? (a) 37 (b) 13 (c) 19 (d) 25
  20. 20.If a + b = 10 and ab = 16, what is a² + b²? (a) 132 (b) 100 (c) 68 (d) 84
  21. 21.What are the roots of x² − 5x + 6 = 0? (a) 2 and 4 (b) 1 and 6 (c) 1 and 4 (d) 2 and 3
  22. 22.If a + b = 18 and ab = 72, what is (a − b)²? (a) 36 (b) 324 (c) 6 (d) 180
  23. 23.If a + b = 17 and ab = 72, what is a² + b²? (a) 289 (b) 433 (c) 145 (d) 217
  24. 24.If a + b = 3 and ab = 2, what is (a − b)²? (a) 5 (b) 7 (c) 9 (d) 1
  25. 25.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 5786 (b) 6384 (c) 5776 (d) 7056
  26. 26.If a + b = 15 and ab = 36, what is a² + b²? (a) 153 (b) 225 (c) 189 (d) 297
  27. 27.If a + b = 11 and ab = 28, what is a² + b²? (a) 65 (b) 93 (c) 177 (d) 121
  28. 28.Which is the correct factorisation of x² + 8x + 12? (a) (x − 2)(x − 6) (b) (x + 2)(x + 6) (c) (x + 1)(x + 7) (d) (x + 1)(x + 12)
  29. 29.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x − 4)(x − 9) (c) (x + 1)(x + 12) (d) (x + 4)(x + 9)
  30. 30.If a + b = 11 and ab = 18, what is a² + b²? (a) 121 (b) 103 (c) 85 (d) 157
  31. 31.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4356 (b) 4916 (c) 5476 (d) 5486
  32. 32.What are the roots of x² − 10x + 24 = 0? (a) 1 and 9 (b) 4 and 6 (c) 1 and 24 (d) 2 and 12
  33. 33.What is the sum of the roots of x² − 12x + 35 = 0? (a) 13 (b) 11 (c) 35 (d) 12
  34. 34.What is the product of the roots of x² − 10x + 21 = 0? (a) 10 (b) 22 (c) 20 (d) 21
  35. 35.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1596 (b) 1764 (c) 1444 (d) 1454
  36. 36.If a + b = 13 and ab = 12, what is (a − b)²? (a) 121 (b) 11 (c) 169 (d) 145
  37. 37.If a + b = 9 and ab = 14, what is (a − b)²? (a) 53 (b) 25 (c) 81 (d) 5
  38. 38.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x + 1)(x + 8) (c) (x + 2)(x + 4) (d) (x − 2)(x − 4)
  39. 39.If a + b = 16 and ab = 60, what is a² + b²? (a) 376 (b) 196 (c) 136 (d) 256
  40. 40.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 576 (c) 416 (d) 586

Answer key

  1. 6 and 7
  2. 185
  3. 36
  4. 7225
  5. (x + 8)(x + 9)
  6. 1521
  7. 9
  8. 2809
  9. 8464
  10. 3844
  11. 8281
  12. 56
  13. (x + 5)(x + 9)
  14. 16
  15. 529
  16. 100
  17. 25
  18. 17
  19. 13
  20. 68
  21. 2 and 3
  22. 36
  23. 145
  24. 1
  25. 5776
  26. 153
  27. 65
  28. (x + 2)(x + 6)
  29. (x + 4)(x + 9)
  30. 85
  31. 5476
  32. 4 and 6
  33. 12
  34. 21
  35. 1444
  36. 121
  37. 25
  38. (x + 2)(x + 4)
  39. 136
  40. 576

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