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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 492 · 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 = 15 and ab = 56, what is a² + b²? (a) 225 (b) 337 (c) 169 (d) 113
  2. 2.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 73 (c) 72 (d) 17
  3. 3.If a + b = 12 and ab = 35, what is a² + b²? (a) 214 (b) 74 (c) 109 (d) 144
  4. 4.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8649 (b) 7579 (c) 7569 (d) 8091
  5. 5.What is the sum of the roots of x² − 6x + 8 = 0? (a) 7 (b) 6 (c) 8 (d) 5
  6. 6.If a + b = 13 and ab = 12, what is (a − b)²? (a) 121 (b) 169 (c) 145 (d) 11
  7. 7.If a + b = 9 and ab = 14, what is (a − b)²? (a) 5 (b) 81 (c) 25 (d) 53
  8. 8.If a + b = 20 and ab = 99, what is a² + b²? (a) 301 (b) 400 (c) 202 (d) 598
  9. 9.If a + b = 15 and ab = 44, what is a² + b²? (a) 313 (b) 137 (c) 181 (d) 225
  10. 10.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x − 8)(x − 9) (c) (x + 8)(x + 9) (d) (x + 1)(x + 16)
  11. 11.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1369 (b) 1849 (c) 1859 (d) 1609
  12. 12.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9035 (b) 9025 (c) 9975 (d) 11025
  13. 13.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 904 (b) 1024 (c) 1034 (d) 784
  14. 14.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4925 (b) 5635 (c) 5625 (d) 4225
  15. 15.What is the sum of the roots of x² − 12x + 32 = 0? (a) 11 (b) 13 (c) 32 (d) 12
  16. 16.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2035 (b) 1225 (c) 2025 (d) 1625
  17. 17.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 1)(x + 48) (c) (x + 6)(x + 8) (d) (x − 6)(x − 8)
  18. 18.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 1024 (b) 784 (c) 896 (d) 794
  19. 19.If a + b = 11 and ab = 18, what is a² + b²? (a) 121 (b) 103 (c) 85 (d) 157
  20. 20.If a + b = 17 and ab = 70, what is a² + b²? (a) 149 (b) 429 (c) 289 (d) 219
  21. 21.If a + b = 8 and ab = 15, what is a² + b²? (a) 49 (b) 34 (c) 94 (d) 64
  22. 22.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1936 (b) 1296 (c) 1584 (d) 1306
  23. 23.If a + b = 14 and ab = 48, what is (a − b)²? (a) 4 (b) 2 (c) 196 (d) 100
  24. 24.What is the product of the roots of x² − 13x + 36 = 0? (a) 35 (b) 36 (c) 13 (d) 37
  25. 25.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 334 (b) 396 (c) 484 (d) 324
  26. 26.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 841 (b) 899 (c) 961 (d) 851
  27. 27.If a + b = 5 and ab = 4, what is a² + b²? (a) 21 (b) 17 (c) 25 (d) 33
  28. 28.Which is the correct factorisation of x² + 13x + 40? (a) (x + 5)(x + 8) (b) (x + 1)(x + 40) (c) (x − 5)(x − 8) (d) (x + 1)(x + 12)
  29. 29.If a + b = 16 and ab = 60, what is (a − b)²? (a) 16 (b) 4 (c) 256 (d) 136
  30. 30.If a + b = 9 and ab = 8, what is a² + b²? (a) 81 (b) 97 (c) 73 (d) 65
  31. 31.If a + b = 13 and ab = 30, what is a² + b²? (a) 109 (b) 169 (c) 229 (d) 139
  32. 32.What is the product of the roots of x² − 10x + 21 = 0? (a) 22 (b) 20 (c) 21 (d) 10
  33. 33.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 235 (b) 225 (c) 375 (d) 625
  34. 34.If a + b = 11 and ab = 18, what is (a − b)²? (a) 85 (b) 49 (c) 121 (d) 7
  35. 35.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 875 (b) 1225 (c) 625 (d) 635
  36. 36.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 333 (c) 225 (d) 657
  37. 37.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 371 (b) 399 (c) 441 (d) 361
  38. 38.What is the sum of the roots of x² − 11x + 30 = 0? (a) 12 (b) 11 (c) 30 (d) 10
  39. 39.If a + b = 12 and ab = 32, what is a² + b²? (a) 144 (b) 112 (c) 208 (d) 80
  40. 40.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x + 2)(x + 8) (c) (x − 2)(x − 8) (d) (x + 1)(x + 16)

Answer key

  1. 113
  2. 72
  3. 74
  4. 7569
  5. 6
  6. 121
  7. 25
  8. 202
  9. 137
  10. (x + 8)(x + 9)
  11. 1849
  12. 9025
  13. 1024
  14. 5625
  15. 12
  16. 2025
  17. (x + 6)(x + 8)
  18. 784
  19. 85
  20. 149
  21. 34
  22. 1296
  23. 4
  24. 36
  25. 324
  26. 841
  27. 17
  28. (x + 5)(x + 8)
  29. 16
  30. 65
  31. 109
  32. 21
  33. 225
  34. 49
  35. 625
  36. 225
  37. 361
  38. 11
  39. 80
  40. (x + 2)(x + 8)

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