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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 883 · 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.If a + b = 16 and ab = 55, what is (a − b)²? (a) 6 (b) 256 (c) 146 (d) 36
  2. 2.If a + b = 12 and ab = 35, what is (a − b)²? (a) 74 (b) 144 (c) 2 (d) 4
  3. 3.If a + b = 17 and ab = 60, what is a² + b²? (a) 289 (b) 409 (c) 169 (d) 229
  4. 4.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 686 (b) 1156 (c) 676 (d) 884
  5. 5.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)
  6. 6.What is the sum of the roots of x² − 10x + 16 = 0? (a) 11 (b) 16 (c) 10 (d) 9
  7. 7.What are the roots of x² − 14x + 45 = 0? (a) 1 and 13 (b) 3 and 15 (c) 1 and 45 (d) 5 and 9
  8. 8.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 5476 (b) 4366 (c) 4884 (d) 4356
  9. 9.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 9216 (b) 10826 (c) 10816 (d) 10016
  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 + 6) (d) (x + 1)(x + 12)
  11. 11.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 396 (b) 324 (c) 484 (d) 334
  12. 12.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 14) (c) (x + 6)(x + 9) (d) (x + 1)(x + 54)
  13. 13.What is the sum of the roots of x² − 7x + 10 = 0? (a) 7 (b) 8 (c) 10 (d) 6
  14. 14.If a + b = 17 and ab = 70, what is a² + b²? (a) 149 (b) 429 (c) 219 (d) 289
  15. 15.If a + b = 20 and ab = 96, what is (a − b)²? (a) 4 (b) 208 (c) 16 (d) 400
  16. 16.If a + b = 8 and ab = 15, what is a² + b²? (a) 34 (b) 94 (c) 49 (d) 64
  17. 17.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x − 2)(x − 7) (c) (x + 2)(x + 7) (d) (x + 1)(x + 8)
  18. 18.What is the sum of the roots of x² − 8x + 15 = 0? (a) 7 (b) 15 (c) 8 (d) 9
  19. 19.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 841 (b) 901 (c) 961 (d) 971
  20. 20.If a + b = 12 and ab = 20, what is (a − b)²? (a) 8 (b) 144 (c) 64 (d) 104
  21. 21.If a + b = 13 and ab = 40, what is a² + b²? (a) 129 (b) 89 (c) 249 (d) 169
  22. 22.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7754 (b) 8096 (c) 8464 (d) 7744
  23. 23.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 494 (b) 404 (c) 484 (d) 324
  24. 24.What is the sum of the roots of x² − 12x + 27 = 0? (a) 11 (b) 13 (c) 27 (d) 12
  25. 25.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 7225 (b) 5625 (c) 5635 (d) 6375
  26. 26.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2401 (b) 2601 (c) 2501 (d) 2611
  27. 27.If a + b = 18 and ab = 80, what is (a − b)²? (a) 324 (b) 164 (c) 2 (d) 4
  28. 28.If a + b = 15 and ab = 56, what is (a − b)²? (a) 113 (b) 1 (c) 169 (d) 225
  29. 29.What is the sum of the roots of x² − 9x + 14 = 0? (a) 14 (b) 8 (c) 10 (d) 9
  30. 30.If a + b = 10 and ab = 21, what is (a − b)²? (a) 4 (b) 58 (c) 16 (d) 100
  31. 31.If a + b = 17 and ab = 72, what is (a − b)²? (a) 1 (b) 217 (c) 289 (d) 145
  32. 32.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6561 (b) 6571 (c) 6241 (d) 6401
  33. 33.What is the sum of the roots of x² − 15x + 54 = 0? (a) 15 (b) 54 (c) 16 (d) 14
  34. 34.What is the product of the roots of x² − 13x + 42 = 0? (a) 43 (b) 42 (c) 13 (d) 41
  35. 35.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8109 (b) 7569 (c) 8659 (d) 8649
  36. 36.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
  37. 37.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3969 (b) 3259 (c) 3249 (d) 3591
  38. 38.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 8125 (b) 9025 (c) 7225 (d) 9035
  39. 39.If a + b = 20 and ab = 99, what is (a − b)²? (a) 4 (b) 202 (c) 400 (d) 2
  40. 40.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1764 (b) 1604 (c) 1444 (d) 1774

Answer key

  1. 36
  2. 4
  3. 169
  4. 676
  5. (x + 2)(x + 5)
  6. 10
  7. 5 and 9
  8. 4356
  9. 10816
  10. (x + 3)(x + 4)
  11. 324
  12. (x + 6)(x + 9)
  13. 7
  14. 149
  15. 16
  16. 34
  17. (x + 2)(x + 7)
  18. 8
  19. 961
  20. 64
  21. 89
  22. 7744
  23. 484
  24. 12
  25. 5625
  26. 2601
  27. 4
  28. 1
  29. 9
  30. 16
  31. 1
  32. 6561
  33. 15
  34. 42
  35. 8649
  36. 7 and 8
  37. 3249
  38. 9025
  39. 4
  40. 1764

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