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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 981 · 40 questions
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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 = 11 and ab = 24, what is (a − b)²? (a) 25 (b) 5 (c) 73 (d) 121
  2. 2.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6889 (b) 5929 (c) 6391 (d) 5939
  3. 3.What are the roots of x² − 14x + 48 = 0? (a) 1 and 13 (b) 6 and 8 (c) 2 and 24 (d) 1 and 48
  4. 4.What is the product of the roots of x² − 12x + 27 = 0? (a) 27 (b) 26 (c) 28 (d) 12
  5. 5.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 576 (b) 586 (c) 416 (d) 256
  6. 6.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8091 (b) 7579 (c) 7569 (d) 8649
  7. 7.If a + b = 12 and ab = 20, what is a² + b²? (a) 144 (b) 104 (c) 184 (d) 124
  8. 8.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x + 1)(x + 13) (c) (x + 1)(x + 48) (d) (x − 6)(x − 8)
  9. 9.If a + b = 12 and ab = 27, what is (a − b)²? (a) 6 (b) 36 (c) 144 (d) 90
  10. 10.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4225 (b) 4925 (c) 5625 (d) 5635
  11. 11.If a + b = 18 and ab = 80, what is (a − b)²? (a) 164 (b) 324 (c) 2 (d) 4
  12. 12.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1591 (b) 1379 (c) 1849 (d) 1369
  13. 13.What is the product of the roots of x² − 9x + 18 = 0? (a) 18 (b) 9 (c) 17 (d) 19
  14. 14.If a + b = 9 and ab = 8, what is (a − b)²? (a) 7 (b) 65 (c) 81 (d) 49
  15. 15.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2601 (b) 2401 (c) 2501 (d) 2611
  16. 16.If a + b = 14 and ab = 48, what is a² + b²? (a) 292 (b) 100 (c) 196 (d) 148
  17. 17.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 5184 (b) 4904 (c) 4624 (d) 5194
  18. 18.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 1)(x + 11) (c) (x + 3)(x + 9) (d) (x + 1)(x + 27)
  19. 19.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
  20. 20.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x − 7)(x − 8) (c) (x + 7)(x + 8) (d) (x + 1)(x + 56)
  21. 21.If a + b = 15 and ab = 44, what is (a − b)²? (a) 137 (b) 225 (c) 7 (d) 49
  22. 22.If a + b = 20 and ab = 99, what is (a − b)²? (a) 4 (b) 2 (c) 202 (d) 400
  23. 23.If a + b = 19 and ab = 84, what is (a − b)²? (a) 193 (b) 361 (c) 25 (d) 5
  24. 24.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x + 4)(x + 9) (c) (x + 1)(x + 36) (d) (x − 4)(x − 9)
  25. 25.If a + b = 18 and ab = 77, what is a² + b²? (a) 170 (b) 324 (c) 247 (d) 478
  26. 26.If a + b = 14 and ab = 48, what is (a − b)²? (a) 196 (b) 2 (c) 4 (d) 100
  27. 27.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2714 (b) 2704 (c) 2504 (d) 2304
  28. 28.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9975 (b) 9035 (c) 11025 (d) 9025
  29. 29.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 1 (c) 361 (d) 271
  30. 30.If a + b = 16 and ab = 48, what is a² + b²? (a) 352 (b) 208 (c) 256 (d) 160
  31. 31.Which is the correct factorisation of x² + 10x + 21? (a) (x − 3)(x − 7) (b) (x + 1)(x + 9) (c) (x + 1)(x + 21) (d) (x + 3)(x + 7)
  32. 32.If a + b = 20 and ab = 96, what is (a − b)²? (a) 4 (b) 16 (c) 400 (d) 208
  33. 33.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 375 (b) 625 (c) 225 (d) 235
  34. 34.If a + b = 7 and ab = 12, what is a² + b²? (a) 73 (b) 37 (c) 49 (d) 25
  35. 35.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3721 (b) 3601 (c) 3481 (d) 3731
  36. 36.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 6)(x + 7) (c) (x + 1)(x + 42) (d) (x − 6)(x − 7)
  37. 37.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6899 (b) 6889 (c) 6409 (d) 5929
  38. 38.If a + b = 17 and ab = 66, what is (a − b)²? (a) 5 (b) 25 (c) 157 (d) 289
  39. 39.If a + b = 15 and ab = 56, what is a² + b²? (a) 337 (b) 225 (c) 113 (d) 169
  40. 40.If a + b = 11 and ab = 30, what is a² + b²? (a) 121 (b) 91 (c) 181 (d) 61

Answer key

  1. 25
  2. 5929
  3. 6 and 8
  4. 27
  5. 576
  6. 7569
  7. 104
  8. (x + 6)(x + 8)
  9. 36
  10. 5625
  11. 4
  12. 1369
  13. 18
  14. 49
  15. 2601
  16. 100
  17. 5184
  18. (x + 3)(x + 9)
  19. (x + 8)(x + 9)
  20. (x + 7)(x + 8)
  21. 49
  22. 4
  23. 25
  24. (x + 4)(x + 9)
  25. 170
  26. 4
  27. 2704
  28. 9025
  29. 1
  30. 160
  31. (x + 3)(x + 7)
  32. 16
  33. 225
  34. 25
  35. 3721
  36. (x + 6)(x + 7)
  37. 6889
  38. 25
  39. 113
  40. 61

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