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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 834 · 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 + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 63) (c) (x + 1)(x + 15) (d) (x − 7)(x − 9)
  2. 2.What is the sum of the roots of x² − 11x + 30 = 0? (a) 10 (b) 11 (c) 30 (d) 12
  3. 3.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 686 (b) 884 (c) 676 (d) 1156
  4. 4.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 202 (c) 400 (d) 4
  5. 5.If a + b = 15 and ab = 44, what is (a − b)²? (a) 225 (b) 49 (c) 137 (d) 7
  6. 6.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 71 (c) 17 (d) 72
  7. 7.If a + b = 17 and ab = 66, what is (a − b)²? (a) 157 (b) 25 (c) 5 (d) 289
  8. 8.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 784 (b) 1024 (c) 904 (d) 1034
  9. 9.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x + 1)(x + 8) (c) (x + 4)(x + 5) (d) (x − 4)(x − 5)
  10. 10.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 3)(x + 4) (c) (x + 1)(x + 12) (d) (x + 1)(x + 6)
  11. 11.What is the sum of the roots of x² − 12x + 27 = 0? (a) 27 (b) 13 (c) 11 (d) 12
  12. 12.If a + b = 18 and ab = 80, what is a² + b²? (a) 244 (b) 324 (c) 484 (d) 164
  13. 13.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 15 (c) 16 (d) 56
  14. 14.What are the roots of x² − 12x + 27 = 0? (a) 3 and 9 (b) 2 and 10 (c) 1 and 27 (d) 1 and 11
  15. 15.What is the sum of the roots of x² − 6x + 8 = 0? (a) 5 (b) 7 (c) 8 (d) 6
  16. 16.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x − 3)(x − 5) (c) (x + 3)(x + 5) (d) (x + 1)(x + 7)
  17. 17.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3721 (b) 3491 (c) 3599 (d) 3481
  18. 18.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x − 7)(x − 8) (c) (x + 1)(x + 56) (d) (x + 1)(x + 14)
  19. 19.If a + b = 11 and ab = 28, what is a² + b²? (a) 121 (b) 93 (c) 177 (d) 65
  20. 20.If a + b = 12 and ab = 32, what is a² + b²? (a) 112 (b) 144 (c) 80 (d) 208
  21. 21.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 484 (b) 396 (c) 324 (d) 334
  22. 22.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4925 (b) 5635 (c) 4225 (d) 5625
  23. 23.If a + b = 11 and ab = 18, what is a² + b²? (a) 85 (b) 121 (c) 103 (d) 157
  24. 24.If a + b = 16 and ab = 60, what is (a − b)²? (a) 136 (b) 4 (c) 256 (d) 16
  25. 25.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 324 (b) 484 (c) 404 (d) 494
  26. 26.If a + b = 10 and ab = 21, what is a² + b²? (a) 142 (b) 79 (c) 100 (d) 58
  27. 27.What are the roots of x² − 16x + 63 = 0? (a) 3 and 21 (b) 1 and 15 (c) 1 and 63 (d) 7 and 9
  28. 28.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1521 (b) 1681 (c) 1691 (d) 1601
  29. 29.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3259 (b) 3969 (c) 3249 (d) 3591
  30. 30.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9419 (b) 9409 (c) 9991 (d) 10609
  31. 31.What is the product of the roots of x² − 9x + 14 = 0? (a) 9 (b) 15 (c) 14 (d) 13
  32. 32.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9035 (b) 9975 (c) 9025 (d) 11025
  33. 33.If a + b = 9 and ab = 20, what is a² + b²? (a) 41 (b) 81 (c) 61 (d) 121
  34. 34.What are the roots of x² − 15x + 56 = 0? (a) 7 and 8 (b) 1 and 14 (c) 2 and 28 (d) 1 and 56
  35. 35.What is the sum of the roots of x² − 15x + 54 = 0? (a) 15 (b) 54 (c) 14 (d) 16
  36. 36.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 16) (b) (x − 2)(x − 8) (c) (x + 2)(x + 8) (d) (x + 1)(x + 9)
  37. 37.If a + b = 14 and ab = 40, what is a² + b²? (a) 156 (b) 116 (c) 196 (d) 276
  38. 38.What are the roots of x² − 5x + 6 = 0? (a) 1 and 4 (b) 1 and 6 (c) 2 and 4 (d) 2 and 3
  39. 39.What is the sum of the roots of x² − 12x + 35 = 0? (a) 35 (b) 12 (c) 13 (d) 11
  40. 40.If a + b = 17 and ab = 72, what is a² + b²? (a) 145 (b) 289 (c) 433 (d) 217

Answer key

  1. (x + 7)(x + 9)
  2. 11
  3. 676
  4. 4
  5. 49
  6. 72
  7. 25
  8. 1024
  9. (x + 4)(x + 5)
  10. (x + 3)(x + 4)
  11. 12
  12. 164
  13. 15
  14. 3 and 9
  15. 6
  16. (x + 3)(x + 5)
  17. 3481
  18. (x + 7)(x + 8)
  19. 65
  20. 80
  21. 324
  22. 5625
  23. 85
  24. 16
  25. 484
  26. 58
  27. 7 and 9
  28. 1681
  29. 3249
  30. 9409
  31. 14
  32. 9025
  33. 41
  34. 7 and 8
  35. 15
  36. (x + 2)(x + 8)
  37. 116
  38. 2 and 3
  39. 12
  40. 145

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