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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 325 · 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 56²? (a) 3146 (b) 3136 (c) 3584 (d) 4096
  2. 2.If a + b = 19 and ab = 84, what is (a − b)²? (a) 193 (b) 361 (c) 5 (d) 25
  3. 3.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 371 (b) 441 (c) 361 (d) 399
  4. 4.If a + b = 4 and ab = 3, what is a² + b²? (a) 16 (b) 10 (c) 22 (d) 13
  5. 5.What is the sum of the roots of x² − 7x + 12 = 0? (a) 7 (b) 8 (c) 12 (d) 6
  6. 6.If a + b = 20 and ab = 96, what is a² + b²? (a) 304 (b) 592 (c) 208 (d) 400
  7. 7.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8464 (b) 8096 (c) 7754 (d) 7744
  8. 8.What is the sum of the roots of x² − 14x + 48 = 0? (a) 13 (b) 15 (c) 48 (d) 14
  9. 9.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 202 (c) 400 (d) 4
  10. 10.If a + b = 15 and ab = 44, what is (a − b)²? (a) 7 (b) 225 (c) 49 (d) 137
  11. 11.If a + b = 8 and ab = 7, what is a² + b²? (a) 64 (b) 50 (c) 78 (d) 57
  12. 12.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3616 (b) 3136 (c) 4106 (d) 4096
  13. 13.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 17 (c) 73 (d) 72
  14. 14.If a + b = 19 and ab = 84, what is a² + b²? (a) 193 (b) 529 (c) 277 (d) 361
  15. 15.If a + b = 11 and ab = 28, what is a² + b²? (a) 65 (b) 93 (c) 177 (d) 121
  16. 16.If a + b = 16 and ab = 55, what is a² + b²? (a) 201 (b) 256 (c) 146 (d) 366
  17. 17.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1849 (b) 1369 (c) 1859 (d) 1609
  18. 18.If a + b = 13 and ab = 22, what is a² + b²? (a) 169 (b) 213 (c) 147 (d) 125
  19. 19.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 1)(x + 45) (c) (x + 1)(x + 13) (d) (x + 5)(x + 9)
  20. 20.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x − 3)(x − 6) (c) (x + 3)(x + 6) (d) (x + 1)(x + 18)
  21. 21.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3025 (b) 3575 (c) 3035 (d) 4225
  22. 22.If a + b = 17 and ab = 70, what is (a − b)²? (a) 3 (b) 9 (c) 289 (d) 149
  23. 23.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 5776 (b) 5786 (c) 6384 (d) 7056
  24. 24.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x − 2)(x − 6) (c) (x + 1)(x + 12) (d) (x + 2)(x + 6)
  25. 25.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 5929 (b) 6391 (c) 6889 (d) 5939
  26. 26.If a + b = 13 and ab = 40, what is (a − b)²? (a) 89 (b) 169 (c) 3 (d) 9
  27. 27.What are the roots of x² − 15x + 54 = 0? (a) 1 and 54 (b) 6 and 9 (c) 2 and 27 (d) 1 and 14
  28. 28.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 904 (b) 1034 (c) 1024 (d) 784
  29. 29.What are the roots of x² − 9x + 20 = 0? (a) 1 and 8 (b) 2 and 10 (c) 4 and 5 (d) 1 and 20
  30. 30.If a + b = 20 and ab = 96, what is (a − b)²? (a) 4 (b) 208 (c) 16 (d) 400
  31. 31.What are the roots of x² − 6x + 8 = 0? (a) 2 and 4 (b) 2 and 5 (c) 1 and 8 (d) 1 and 5
  32. 32.If a + b = 21 and ab = 108, what is a² + b²? (a) 657 (b) 333 (c) 441 (d) 225
  33. 33.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 576 (b) 416 (c) 586 (d) 256
  34. 34.If a + b = 13 and ab = 30, what is (a − b)²? (a) 169 (b) 7 (c) 49 (d) 109
  35. 35.If a + b = 17 and ab = 60, what is a² + b²? (a) 409 (b) 229 (c) 289 (d) 169
  36. 36.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 18) (b) (x − 2)(x − 9) (c) (x + 2)(x + 9) (d) (x + 1)(x + 10)
  37. 37.What are the roots of x² − 12x + 35 = 0? (a) 1 and 35 (b) 2 and 10 (c) 1 and 11 (d) 5 and 7
  38. 38.What is the sum of the roots of x² − 11x + 24 = 0? (a) 24 (b) 10 (c) 11 (d) 12
  39. 39.What is the sum of the roots of x² − 11x + 28 = 0? (a) 10 (b) 28 (c) 12 (d) 11
  40. 40.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)

Answer key

  1. 3136
  2. 25
  3. 361
  4. 10
  5. 7
  6. 208
  7. 7744
  8. 14
  9. 4
  10. 49
  11. 50
  12. 4096
  13. 72
  14. 193
  15. 65
  16. 146
  17. 1849
  18. 125
  19. (x + 5)(x + 9)
  20. (x + 3)(x + 6)
  21. 3025
  22. 9
  23. 5776
  24. (x + 2)(x + 6)
  25. 5929
  26. 9
  27. 6 and 9
  28. 1024
  29. 4 and 5
  30. 16
  31. 2 and 4
  32. 225
  33. 576
  34. 49
  35. 169
  36. (x + 2)(x + 9)
  37. 5 and 7
  38. 11
  39. 11
  40. (x + 8)(x + 9)

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