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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 121 · 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.What are the roots of x² − 7x + 10 = 0? (a) 1 and 6 (b) 1 and 10 (c) 3 and 4 (d) 2 and 5
  2. 2.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 15 (c) 56 (d) 16
  3. 3.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8649 (b) 8659 (c) 7569 (d) 8109
  4. 4.If a + b = 9 and ab = 14, what is a² + b²? (a) 81 (b) 67 (c) 109 (d) 53
  5. 5.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 400 (c) 301 (d) 202
  6. 6.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 2)(x + 4) (c) (x + 1)(x + 8) (d) (x + 1)(x + 5)
  7. 7.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7225 (b) 8075 (c) 7235 (d) 9025
  8. 8.If a + b = 8 and ab = 7, what is a² + b²? (a) 57 (b) 50 (c) 64 (d) 78
  9. 9.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 3)(x + 9) (c) (x + 1)(x + 11) (d) (x + 1)(x + 27)
  10. 10.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6241 (b) 6571 (c) 6401 (d) 6561
  11. 11.What are the roots of x² − 15x + 56 = 0? (a) 7 and 8 (b) 2 and 28 (c) 1 and 56 (d) 1 and 14
  12. 12.If a + b = 19 and ab = 84, what is (a − b)²? (a) 25 (b) 361 (c) 5 (d) 193
  13. 13.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 529 (b) 299 (c) 289 (d) 391
  14. 14.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 9 (c) 225 (d) 3
  15. 15.What is the sum of the roots of x² − 9x + 20 = 0? (a) 20 (b) 8 (c) 10 (d) 9
  16. 16.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 11) (b) (x + 4)(x + 8) (c) (x − 4)(x − 8) (d) (x + 1)(x + 32)
  17. 17.What is the sum of the roots of x² − 12x + 27 = 0? (a) 27 (b) 13 (c) 11 (d) 12
  18. 18.If a + b = 16 and ab = 48, what is a² + b²? (a) 160 (b) 256 (c) 352 (d) 208
  19. 19.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 1)(x + 14) (c) (x − 6)(x − 9) (d) (x + 6)(x + 9)
  20. 20.If a + b = 5 and ab = 6, what is (a − b)²? (a) 13 (b) 19 (c) 25 (d) 1
  21. 21.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1225 (b) 2025 (c) 1235 (d) 1575
  22. 22.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 225 (c) 657 (d) 333
  23. 23.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 5625 (b) 5635 (c) 4925 (d) 4225
  24. 24.If a + b = 10 and ab = 21, what is a² + b²? (a) 58 (b) 142 (c) 100 (d) 79
  25. 25.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x + 3)(x + 8) (c) (x + 1)(x + 24) (d) (x − 3)(x − 8)
  26. 26.If a + b = 19 and ab = 88, what is (a − b)²? (a) 361 (b) 9 (c) 185 (d) 3
  27. 27.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 63 (c) 15 (d) 17
  28. 28.If a + b = 10 and ab = 21, what is (a − b)²? (a) 16 (b) 58 (c) 4 (d) 100
  29. 29.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4771 (b) 4899 (c) 5041 (d) 4761
  30. 30.If a + b = 19 and ab = 88, what is a² + b²? (a) 537 (b) 361 (c) 273 (d) 185
  31. 31.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1296 (b) 1616 (c) 1936 (d) 1946
  32. 32.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7066 (b) 6416 (c) 5776 (d) 7056
  33. 33.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 14) (c) (x + 1)(x + 56) (d) (x − 7)(x − 8)
  34. 34.If a + b = 12 and ab = 20, what is a² + b²? (a) 184 (b) 104 (c) 144 (d) 124
  35. 35.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8096 (b) 8464 (c) 7754 (d) 7744
  36. 36.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 16) (c) (x + 1)(x + 72) (d) (x − 8)(x − 9)
  37. 37.What is the product of the roots of x² − 11x + 30 = 0? (a) 30 (b) 11 (c) 29 (d) 31
  38. 38.If a + b = 15 and ab = 36, what is (a − b)²? (a) 153 (b) 81 (c) 9 (d) 225
  39. 39.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x − 7)(x − 9) (c) (x + 1)(x + 63) (d) (x + 1)(x + 15)
  40. 40.If a + b = 13 and ab = 30, what is a² + b²? (a) 229 (b) 139 (c) 109 (d) 169

Answer key

  1. 2 and 5
  2. 15
  3. 8649
  4. 53
  5. 202
  6. (x + 2)(x + 4)
  7. 7225
  8. 50
  9. (x + 3)(x + 9)
  10. 6561
  11. 7 and 8
  12. 25
  13. 289
  14. 9
  15. 9
  16. (x + 4)(x + 8)
  17. 12
  18. 160
  19. (x + 6)(x + 9)
  20. 1
  21. 1225
  22. 225
  23. 5625
  24. 58
  25. (x + 3)(x + 8)
  26. 9
  27. 16
  28. 16
  29. 4761
  30. 185
  31. 1936
  32. 7056
  33. (x + 7)(x + 8)
  34. 104
  35. 7744
  36. (x + 8)(x + 9)
  37. 30
  38. 81
  39. (x + 7)(x + 9)
  40. 109

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