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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 117 · 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 75²? (a) 4225 (b) 5625 (c) 4925 (d) 5635
  2. 2.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 289 (b) 299 (c) 391 (d) 529
  3. 3.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 1 (c) 271 (d) 361
  4. 4.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 1)(x + 8) (c) (x + 2)(x + 4) (d) (x + 1)(x + 5)
  5. 5.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2601 (b) 2611 (c) 2401 (d) 2501
  6. 6.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)
  7. 7.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4899 (b) 4771 (c) 4761 (d) 5041
  8. 8.If a + b = 10 and ab = 16, what is (a − b)²? (a) 6 (b) 100 (c) 36 (d) 68
  9. 9.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6561 (b) 6399 (c) 6241 (d) 6251
  10. 10.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4356 (b) 5476 (c) 4366 (d) 4884
  11. 11.If a + b = 7 and ab = 12, what is (a − b)²? (a) 49 (b) 25 (c) 1 (d) 37
  12. 12.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 72 (c) 17 (d) 16
  13. 13.If a + b = 11 and ab = 18, what is (a − b)²? (a) 85 (b) 49 (c) 121 (d) 7
  14. 14.If a + b = 19 and ab = 88, what is (a − b)²? (a) 185 (b) 9 (c) 361 (d) 3
  15. 15.What is the sum of the roots of x² − 8x + 12 = 0? (a) 7 (b) 9 (c) 8 (d) 12
  16. 16.If a + b = 17 and ab = 72, what is (a − b)²? (a) 1 (b) 217 (c) 289 (d) 145
  17. 17.What are the roots of x² − 13x + 42 = 0? (a) 1 and 42 (b) 1 and 12 (c) 6 and 7 (d) 2 and 21
  18. 18.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 71 (c) 73 (d) 17
  19. 19.If a + b = 12 and ab = 11, what is a² + b²? (a) 166 (b) 144 (c) 133 (d) 122
  20. 20.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 18) (b) (x + 1)(x + 10) (c) (x − 2)(x − 9) (d) (x + 2)(x + 9)
  21. 21.If a + b = 17 and ab = 70, what is a² + b²? (a) 149 (b) 289 (c) 219 (d) 429
  22. 22.Which is the correct factorisation of x² + 12x + 32? (a) (x + 4)(x + 8) (b) (x − 4)(x − 8) (c) (x + 1)(x + 32) (d) (x + 1)(x + 11)
  23. 23.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)
  24. 24.If a + b = 6 and ab = 5, what is (a − b)²? (a) 4 (b) 36 (c) 26 (d) 16
  25. 25.What is the product of the roots of x² − 15x + 54 = 0? (a) 55 (b) 53 (c) 54 (d) 15
  26. 26.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1859 (b) 1369 (c) 1849 (d) 1609
  27. 27.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 1156 (b) 884 (c) 676 (d) 686
  28. 28.What is the product of the roots of x² − 15x + 56 = 0? (a) 56 (b) 15 (c) 55 (d) 57
  29. 29.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x − 8)(x − 9)
  30. 30.What is the product of the roots of x² − 8x + 15 = 0? (a) 8 (b) 14 (c) 15 (d) 16
  31. 31.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 6)(x + 9) (c) (x − 6)(x − 9) (d) (x + 1)(x + 14)
  32. 32.If a + b = 13 and ab = 12, what is a² + b²? (a) 157 (b) 193 (c) 169 (d) 145
  33. 33.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 401 (b) 361 (c) 451 (d) 441
  34. 34.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x + 1)(x + 10) (c) (x + 3)(x + 8) (d) (x − 3)(x − 8)
  35. 35.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 14 (c) 15 (d) 16
  36. 36.What are the roots of x² − 13x + 40 = 0? (a) 5 and 8 (b) 1 and 12 (c) 1 and 40 (d) 2 and 20
  37. 37.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4235 (b) 4225 (c) 5625 (d) 4875
  38. 38.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6084 (b) 6724 (c) 6396 (d) 6094
  39. 39.What are the roots of x² − 11x + 30 = 0? (a) 2 and 15 (b) 1 and 10 (c) 5 and 6 (d) 1 and 30
  40. 40.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 361 (b) 371 (c) 399 (d) 441

Answer key

  1. 5625
  2. 289
  3. 1
  4. (x + 2)(x + 4)
  5. 2601
  6. (x + 3)(x + 4)
  7. 4761
  8. 36
  9. 6241
  10. 4356
  11. 1
  12. 17
  13. 49
  14. 9
  15. 8
  16. 1
  17. 6 and 7
  18. 72
  19. 122
  20. (x + 2)(x + 9)
  21. 149
  22. (x + 4)(x + 8)
  23. (x + 6)(x + 8)
  24. 16
  25. 54
  26. 1849
  27. 676
  28. 56
  29. (x + 8)(x + 9)
  30. 15
  31. (x + 6)(x + 9)
  32. 145
  33. 441
  34. (x + 3)(x + 8)
  35. 15
  36. 5 and 8
  37. 4225
  38. 6084
  39. 5 and 6
  40. 361

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