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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 211 · 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.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 1)(x + 24) (c) (x + 4)(x + 6) (d) (x + 1)(x + 9)
  2. 2.If a + b = 8 and ab = 15, what is a² + b²? (a) 94 (b) 49 (c) 64 (d) 34
  3. 3.If a + b = 18 and ab = 77, what is a² + b²? (a) 324 (b) 478 (c) 170 (d) 247
  4. 4.If a + b = 16 and ab = 60, what is a² + b²? (a) 196 (b) 256 (c) 376 (d) 136
  5. 5.What are the roots of x² − 7x + 12 = 0? (a) 1 and 6 (b) 2 and 6 (c) 1 and 12 (d) 3 and 4
  6. 6.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 5625 (b) 4235 (c) 4225 (d) 4875
  7. 7.If a + b = 15 and ab = 50, what is a² + b²? (a) 325 (b) 175 (c) 125 (d) 225
  8. 8.What is the sum of the roots of x² − 11x + 18 = 0? (a) 11 (b) 12 (c) 18 (d) 10
  9. 9.What is the sum of the roots of x² − 11x + 24 = 0? (a) 10 (b) 12 (c) 11 (d) 24
  10. 10.If a + b = 9 and ab = 20, what is a² + b²? (a) 121 (b) 81 (c) 41 (d) 61
  11. 11.If a + b = 13 and ab = 40, what is (a − b)²? (a) 169 (b) 9 (c) 3 (d) 89
  12. 12.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 739 (b) 891 (c) 729 (d) 1089
  13. 13.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 657 (c) 225 (d) 441
  14. 14.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 3035 (b) 2025 (c) 3025 (d) 2525
  15. 15.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8099 (b) 7931 (c) 8281 (d) 7921
  16. 16.Which is the correct factorisation of x² + 13x + 40? (a) (x + 5)(x + 8) (b) (x + 1)(x + 40) (c) (x − 5)(x − 8) (d) (x + 1)(x + 12)
  17. 17.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1444 (b) 1454 (c) 1764 (d) 1596
  18. 18.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 1)(x + 48) (c) (x + 6)(x + 8) (d) (x − 6)(x − 8)
  19. 19.If a + b = 18 and ab = 80, what is a² + b²? (a) 324 (b) 484 (c) 244 (d) 164
  20. 20.What are the roots of x² − 9x + 14 = 0? (a) 2 and 7 (b) 1 and 8 (c) 3 and 6 (d) 1 and 14
  21. 21.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6404 (b) 6084 (c) 6734 (d) 6724
  22. 22.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 529 (b) 391 (c) 289 (d) 299
  23. 23.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 441 (c) 9 (d) 3
  24. 24.If a + b = 20 and ab = 96, what is a² + b²? (a) 592 (b) 304 (c) 400 (d) 208
  25. 25.If a + b = 11 and ab = 18, what is (a − b)²? (a) 7 (b) 121 (c) 85 (d) 49
  26. 26.If a + b = 14 and ab = 45, what is a² + b²? (a) 286 (b) 196 (c) 151 (d) 106
  27. 27.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 1 and 72 (c) 8 and 9 (d) 1 and 16
  28. 28.What is the product of the roots of x² − 12x + 32 = 0? (a) 32 (b) 31 (c) 33 (d) 12
  29. 29.What is the sum of the roots of x² − 15x + 54 = 0? (a) 14 (b) 15 (c) 16 (d) 54
  30. 30.What are the roots of x² − 10x + 24 = 0? (a) 4 and 6 (b) 1 and 9 (c) 1 and 24 (d) 2 and 12
  31. 31.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)
  32. 32.If a + b = 11 and ab = 18, what is a² + b²? (a) 157 (b) 121 (c) 103 (d) 85
  33. 33.What is the product of the roots of x² − 12x + 27 = 0? (a) 26 (b) 12 (c) 28 (d) 27
  34. 34.What is the product of the roots of x² − 14x + 48 = 0? (a) 14 (b) 49 (c) 48 (d) 47
  35. 35.What is the product of the roots of x² − 15x + 54 = 0? (a) 15 (b) 54 (c) 53 (d) 55
  36. 36.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 1)(x + 11) (c) (x + 5)(x + 7) (d) (x + 1)(x + 35)
  37. 37.What is the product of the roots of x² − 11x + 30 = 0? (a) 29 (b) 11 (c) 30 (d) 31
  38. 38.What are the roots of x² − 7x + 10 = 0? (a) 3 and 4 (b) 1 and 10 (c) 1 and 6 (d) 2 and 5
  39. 39.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 6375 (b) 5625 (c) 7225 (d) 5635
  40. 40.What are the roots of x² − 15x + 54 = 0? (a) 6 and 9 (b) 1 and 14 (c) 2 and 27 (d) 1 and 54

Answer key

  1. (x + 4)(x + 6)
  2. 34
  3. 170
  4. 136
  5. 3 and 4
  6. 4225
  7. 125
  8. 11
  9. 11
  10. 41
  11. 9
  12. 729
  13. 225
  14. 3025
  15. 7921
  16. (x + 5)(x + 8)
  17. 1444
  18. (x + 6)(x + 8)
  19. 164
  20. 2 and 7
  21. 6724
  22. 289
  23. 9
  24. 208
  25. 49
  26. 106
  27. 8 and 9
  28. 32
  29. 15
  30. 4 and 6
  31. (x + 8)(x + 9)
  32. 85
  33. 27
  34. 48
  35. 54
  36. (x + 5)(x + 7)
  37. 30
  38. 2 and 5
  39. 5625
  40. 6 and 9

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