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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 188 · 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 is the sum of the roots of x² − 10x + 21 = 0? (a) 11 (b) 9 (c) 21 (d) 10
  2. 2.What is the sum of the roots of x² − 8x + 15 = 0? (a) 9 (b) 7 (c) 15 (d) 8
  3. 3.If a + b = 18 and ab = 80, what is (a − b)²? (a) 324 (b) 2 (c) 4 (d) 164
  4. 4.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1936 (b) 1296 (c) 1306 (d) 1584
  5. 5.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)
  6. 6.What is the product of the roots of x² − 12x + 27 = 0? (a) 27 (b) 12 (c) 28 (d) 26
  7. 7.What is the sum of the roots of x² − 13x + 36 = 0? (a) 36 (b) 14 (c) 12 (d) 13
  8. 8.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 56 (c) 14 (d) 16
  9. 9.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4875 (b) 5625 (c) 4225 (d) 4235
  10. 10.If a + b = 9 and ab = 18, what is (a − b)²? (a) 45 (b) 81 (c) 9 (d) 3
  11. 11.What is the product of the roots of x² − 16x + 63 = 0? (a) 62 (b) 16 (c) 63 (d) 64
  12. 12.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 1024 (b) 794 (c) 896 (d) 784
  13. 13.If a + b = 11 and ab = 18, what is a² + b²? (a) 121 (b) 157 (c) 103 (d) 85
  14. 14.What is the sum of the roots of x² − 9x + 20 = 0? (a) 10 (b) 8 (c) 20 (d) 9
  15. 15.If a + b = 15 and ab = 56, what is a² + b²? (a) 169 (b) 113 (c) 337 (d) 225
  16. 16.If a + b = 14 and ab = 45, what is a² + b²? (a) 286 (b) 106 (c) 151 (d) 196
  17. 17.If a + b = 8 and ab = 15, what is (a − b)²? (a) 64 (b) 4 (c) 34 (d) 2
  18. 18.What is the product of the roots of x² − 14x + 45 = 0? (a) 45 (b) 44 (c) 46 (d) 14
  19. 19.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 11035 (b) 10025 (c) 9025 (d) 11025
  20. 20.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6396 (b) 6094 (c) 6724 (d) 6084
  21. 21.What is the product of the roots of x² − 13x + 40 = 0? (a) 13 (b) 41 (c) 39 (d) 40
  22. 22.What is the product of the roots of x² − 15x + 56 = 0? (a) 56 (b) 55 (c) 57 (d) 15
  23. 23.If a + b = 18 and ab = 77, what is a² + b²? (a) 170 (b) 247 (c) 324 (d) 478
  24. 24.What are the roots of x² − 12x + 35 = 0? (a) 2 and 10 (b) 5 and 7 (c) 1 and 35 (d) 1 and 11
  25. 25.If a + b = 12 and ab = 27, what is a² + b²? (a) 144 (b) 90 (c) 198 (d) 117
  26. 26.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 409 (b) 529 (c) 539 (d) 289
  27. 27.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 334 (b) 484 (c) 324 (d) 396
  28. 28.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 63) (c) (x + 7)(x + 9) (d) (x + 1)(x + 15)
  29. 29.If a + b = 4 and ab = 3, what is a² + b²? (a) 22 (b) 13 (c) 16 (d) 10
  30. 30.If a + b = 14 and ab = 33, what is a² + b²? (a) 130 (b) 196 (c) 163 (d) 262
  31. 31.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 1)(x + 18) (c) (x + 3)(x + 6) (d) (x + 1)(x + 8)
  32. 32.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4904 (b) 5184 (c) 5194 (d) 4624
  33. 33.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3481 (b) 3599 (c) 3721 (d) 3491
  34. 34.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7056 (b) 7066 (c) 5776 (d) 6416
  35. 35.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 5339 (b) 5329 (c) 4909 (d) 4489
  36. 36.If a + b = 9 and ab = 20, what is a² + b²? (a) 41 (b) 81 (c) 121 (d) 61
  37. 37.If a + b = 11 and ab = 30, what is a² + b²? (a) 91 (b) 61 (c) 121 (d) 181
  38. 38.If a + b = 18 and ab = 80, what is a² + b²? (a) 484 (b) 244 (c) 164 (d) 324
  39. 39.What is the sum of the roots of x² − 12x + 32 = 0? (a) 12 (b) 13 (c) 32 (d) 11
  40. 40.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 6)(x + 8) (c) (x − 6)(x − 8) (d) (x + 1)(x + 48)

Answer key

  1. 10
  2. 8
  3. 4
  4. 1296
  5. (x + 3)(x + 4)
  6. 27
  7. 13
  8. 15
  9. 4225
  10. 9
  11. 63
  12. 784
  13. 85
  14. 9
  15. 113
  16. 106
  17. 4
  18. 45
  19. 11025
  20. 6084
  21. 40
  22. 56
  23. 170
  24. 5 and 7
  25. 90
  26. 529
  27. 324
  28. (x + 7)(x + 9)
  29. 10
  30. 130
  31. (x + 3)(x + 6)
  32. 5184
  33. 3481
  34. 7056
  35. 5329
  36. 41
  37. 61
  38. 164
  39. 12
  40. (x + 6)(x + 8)

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