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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 839 · 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.If a + b = 9 and ab = 18, what is a² + b²? (a) 117 (b) 45 (c) 63 (d) 81
  2. 2.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 324 (b) 484 (c) 494 (d) 404
  3. 3.What are the roots of x² − 10x + 24 = 0? (a) 4 and 6 (b) 2 and 12 (c) 1 and 9 (d) 1 and 24
  4. 4.What is the sum of the roots of x² − 7x + 10 = 0? (a) 6 (b) 10 (c) 7 (d) 8
  5. 5.What is the sum of the roots of x² − 11x + 24 = 0? (a) 12 (b) 10 (c) 24 (d) 11
  6. 6.What is the product of the roots of x² − 9x + 14 = 0? (a) 15 (b) 13 (c) 9 (d) 14
  7. 7.If a + b = 10 and ab = 9, what is (a − b)²? (a) 8 (b) 82 (c) 64 (d) 100
  8. 8.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x + 1)(x + 6) (c) (x + 2)(x + 5) (d) (x − 2)(x − 5)
  9. 9.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 7406 (b) 8836 (c) 7396 (d) 8084
  10. 10.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2304 (b) 2704 (c) 2714 (d) 2504
  11. 11.If a + b = 6 and ab = 8, what is (a − b)²? (a) 20 (b) 4 (c) 36 (d) 2
  12. 12.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4634 (b) 4896 (c) 4624 (d) 5184
  13. 13.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4356 (b) 5486 (c) 4916 (d) 5476
  14. 14.What are the roots of x² − 8x + 15 = 0? (a) 1 and 15 (b) 1 and 7 (c) 2 and 6 (d) 3 and 5
  15. 15.If a + b = 14 and ab = 40, what is (a − b)²? (a) 196 (b) 36 (c) 116 (d) 6
  16. 16.If a + b = 15 and ab = 50, what is (a − b)²? (a) 25 (b) 125 (c) 225 (d) 5
  17. 17.What are the roots of x² − 11x + 30 = 0? (a) 1 and 10 (b) 2 and 15 (c) 1 and 30 (d) 5 and 6
  18. 18.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 266 (b) 576 (c) 384 (d) 256
  19. 19.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x − 8)(x − 9) (c) (x + 8)(x + 9) (d) (x + 1)(x + 16)
  20. 20.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 729 (b) 1089 (c) 909 (d) 1099
  21. 21.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 14) (c) (x + 7)(x + 8) (d) (x + 1)(x + 56)
  22. 22.What is the product of the roots of x² − 16x + 63 = 0? (a) 62 (b) 64 (c) 63 (d) 16
  23. 23.If a + b = 15 and ab = 44, what is (a − b)²? (a) 7 (b) 137 (c) 49 (d) 225
  24. 24.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1946 (b) 1616 (c) 1296 (d) 1936
  25. 25.If a + b = 20 and ab = 99, what is (a − b)²? (a) 4 (b) 202 (c) 2 (d) 400
  26. 26.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2501 (b) 2601 (c) 2611 (d) 2401
  27. 27.What are the roots of x² − 10x + 21 = 0? (a) 1 and 9 (b) 3 and 7 (c) 2 and 8 (d) 1 and 21
  28. 28.If a + b = 15 and ab = 54, what is (a − b)²? (a) 9 (b) 225 (c) 3 (d) 117
  29. 29.If a + b = 15 and ab = 36, what is a² + b²? (a) 297 (b) 189 (c) 225 (d) 153
  30. 30.What are the roots of x² − 13x + 42 = 0? (a) 1 and 12 (b) 1 and 42 (c) 2 and 21 (d) 6 and 7
  31. 31.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 361 (b) 371 (c) 441 (d) 399
  32. 32.What is the sum of the roots of x² − 12x + 27 = 0? (a) 11 (b) 27 (c) 13 (d) 12
  33. 33.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4899 (b) 5041 (c) 4761 (d) 4771
  34. 34.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 391 (b) 529 (c) 289 (d) 299
  35. 35.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7235 (b) 5625 (c) 7225 (d) 6425
  36. 36.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x − 4)(x − 5) (c) (x + 1)(x + 20) (d) (x + 4)(x + 5)
  37. 37.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8291 (b) 8101 (c) 8281 (d) 7921
  38. 38.If a + b = 8 and ab = 7, what is a² + b²? (a) 78 (b) 57 (c) 50 (d) 64
  39. 39.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1691 (b) 1681 (c) 1601 (d) 1521
  40. 40.If a + b = 3 and ab = 2, what is a² + b²? (a) 7 (b) 5 (c) 13 (d) 9

Answer key

  1. 45
  2. 484
  3. 4 and 6
  4. 7
  5. 11
  6. 14
  7. 64
  8. (x + 2)(x + 5)
  9. 7396
  10. 2704
  11. 4
  12. 4624
  13. 5476
  14. 3 and 5
  15. 36
  16. 25
  17. 5 and 6
  18. 256
  19. (x + 8)(x + 9)
  20. 1089
  21. (x + 7)(x + 8)
  22. 63
  23. 49
  24. 1936
  25. 4
  26. 2601
  27. 3 and 7
  28. 9
  29. 153
  30. 6 and 7
  31. 361
  32. 12
  33. 4761
  34. 289
  35. 7225
  36. (x + 4)(x + 5)
  37. 8281
  38. 50
  39. 1681
  40. 5

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