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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 637 · 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.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 266 (b) 384 (c) 256 (d) 576
  2. 2.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3364 (b) 3844 (c) 3374 (d) 3596
  3. 3.If a + b = 19 and ab = 88, what is a² + b²? (a) 273 (b) 537 (c) 185 (d) 361
  4. 4.If a + b = 17 and ab = 72, what is (a − b)²? (a) 145 (b) 289 (c) 1 (d) 217
  5. 5.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 586 (c) 416 (d) 576
  6. 6.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)
  7. 7.If a + b = 6 and ab = 5, what is a² + b²? (a) 46 (b) 36 (c) 26 (d) 31
  8. 8.What is the sum of the roots of x² − 7x + 12 = 0? (a) 8 (b) 7 (c) 6 (d) 12
  9. 9.Which is the correct factorisation of x² + 12x + 27? (a) (x + 3)(x + 9) (b) (x + 1)(x + 27) (c) (x − 3)(x − 9) (d) (x + 1)(x + 11)
  10. 10.If a + b = 5 and ab = 6, what is (a − b)²? (a) 1 (b) 25 (c) 13 (d) 19
  11. 11.What is the product of the roots of x² − 14x + 48 = 0? (a) 48 (b) 14 (c) 47 (d) 49
  12. 12.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 15) (c) (x + 1)(x + 63) (d) (x − 7)(x − 9)
  13. 13.What is the sum of the roots of x² − 11x + 18 = 0? (a) 12 (b) 18 (c) 11 (d) 10
  14. 14.If a + b = 16 and ab = 60, what is (a − b)²? (a) 4 (b) 136 (c) 256 (d) 16
  15. 15.If a + b = 10 and ab = 24, what is (a − b)²? (a) 2 (b) 4 (c) 52 (d) 100
  16. 16.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x + 1)(x + 12) (c) (x − 3)(x − 4) (d) (x + 3)(x + 4)
  17. 17.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x − 3)(x − 8) (c) (x + 1)(x + 10) (d) (x + 3)(x + 8)
  18. 18.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 2)(x + 7) (c) (x + 1)(x + 14) (d) (x + 1)(x + 8)
  19. 19.If a + b = 17 and ab = 66, what is a² + b²? (a) 289 (b) 157 (c) 421 (d) 223
  20. 20.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 494 (b) 404 (c) 484 (d) 324
  21. 21.If a + b = 11 and ab = 24, what is a² + b²? (a) 97 (b) 121 (c) 73 (d) 169
  22. 22.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)
  23. 23.If a + b = 20 and ab = 96, what is a² + b²? (a) 400 (b) 592 (c) 208 (d) 304
  24. 24.What is the product of the roots of x² − 8x + 15 = 0? (a) 15 (b) 8 (c) 14 (d) 16
  25. 25.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4356 (b) 4366 (c) 4884 (d) 5476
  26. 26.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)
  27. 27.If a + b = 7 and ab = 12, what is (a − b)²? (a) 37 (b) 1 (c) 49 (d) 25
  28. 28.What is the product of the roots of x² − 13x + 42 = 0? (a) 43 (b) 42 (c) 13 (d) 41
  29. 29.If a + b = 7 and ab = 6, what is (a − b)²? (a) 37 (b) 49 (c) 5 (d) 25
  30. 30.If a + b = 13 and ab = 40, what is a² + b²? (a) 129 (b) 89 (c) 249 (d) 169
  31. 31.If a + b = 3 and ab = 2, what is (a − b)²? (a) 9 (b) 7 (c) 5 (d) 1
  32. 32.If a + b = 8 and ab = 12, what is (a − b)²? (a) 64 (b) 40 (c) 16 (d) 4
  33. 33.If a + b = 18 and ab = 72, what is (a − b)²? (a) 36 (b) 324 (c) 180 (d) 6
  34. 34.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 5929 (b) 6899 (c) 6409 (d) 6889
  35. 35.What are the roots of x² − 13x + 42 = 0? (a) 2 and 21 (b) 6 and 7 (c) 1 and 42 (d) 1 and 12
  36. 36.What is the sum of the roots of x² − 11x + 24 = 0? (a) 24 (b) 12 (c) 11 (d) 10
  37. 37.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 9025 (b) 7225 (c) 8125 (d) 9035
  38. 38.Which is the correct factorisation of x² + 6x + 8? (a) (x + 2)(x + 4) (b) (x − 2)(x − 4) (c) (x + 1)(x + 5) (d) (x + 1)(x + 8)
  39. 39.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3025 (b) 4235 (c) 4225 (d) 3625
  40. 40.If a + b = 21 and ab = 108, what is a² + b²? (a) 225 (b) 333 (c) 657 (d) 441

Answer key

  1. 256
  2. 3364
  3. 185
  4. 1
  5. 576
  6. (x + 8)(x + 9)
  7. 26
  8. 7
  9. (x + 3)(x + 9)
  10. 1
  11. 48
  12. (x + 7)(x + 9)
  13. 11
  14. 16
  15. 4
  16. (x + 3)(x + 4)
  17. (x + 3)(x + 8)
  18. (x + 2)(x + 7)
  19. 157
  20. 484
  21. 73
  22. (x + 7)(x + 8)
  23. 208
  24. 15
  25. 4356
  26. (x + 2)(x + 9)
  27. 1
  28. 42
  29. 25
  30. 89
  31. 1
  32. 16
  33. 36
  34. 6889
  35. 6 and 7
  36. 11
  37. 9025
  38. (x + 2)(x + 4)
  39. 4225
  40. 225

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