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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 679 · 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 = 6 and ab = 5, what is (a − b)²? (a) 36 (b) 16 (c) 4 (d) 26
  2. 2.If a + b = 4 and ab = 3, what is a² + b²? (a) 13 (b) 16 (c) 10 (d) 22
  3. 3.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 851 (b) 961 (c) 841 (d) 899
  4. 4.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 299 (b) 529 (c) 391 (d) 289
  5. 5.If a + b = 10 and ab = 9, what is (a − b)²? (a) 8 (b) 64 (c) 82 (d) 100
  6. 6.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 12) (c) (x + 1)(x + 6) (d) (x + 3)(x + 4)
  7. 7.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 841 (b) 961 (c) 901 (d) 971
  8. 8.What is the product of the roots of x² − 14x + 48 = 0? (a) 47 (b) 49 (c) 14 (d) 48
  9. 9.If a + b = 17 and ab = 60, what is a² + b²? (a) 169 (b) 289 (c) 409 (d) 229
  10. 10.If a + b = 19 and ab = 90, what is a² + b²? (a) 181 (b) 361 (c) 541 (d) 271
  11. 11.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)
  12. 12.If a + b = 18 and ab = 80, what is a² + b²? (a) 164 (b) 324 (c) 244 (d) 484
  13. 13.If a + b = 13 and ab = 42, what is (a − b)²? (a) 85 (b) 127 (c) 169 (d) 1
  14. 14.If a + b = 13 and ab = 22, what is a² + b²? (a) 169 (b) 213 (c) 147 (d) 125
  15. 15.What are the roots of x² − 11x + 30 = 0? (a) 1 and 10 (b) 2 and 15 (c) 5 and 6 (d) 1 and 30
  16. 16.What is the product of the roots of x² − 9x + 18 = 0? (a) 18 (b) 19 (c) 17 (d) 9
  17. 17.What is the product of the roots of x² − 16x + 63 = 0? (a) 63 (b) 62 (c) 16 (d) 64
  18. 18.If a + b = 9 and ab = 8, what is (a − b)²? (a) 65 (b) 49 (c) 7 (d) 81
  19. 19.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 401 (b) 361 (c) 441 (d) 451
  20. 20.What are the roots of x² − 7x + 10 = 0? (a) 3 and 4 (b) 2 and 5 (c) 1 and 10 (d) 1 and 6
  21. 21.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 54) (c) (x + 6)(x + 9) (d) (x + 1)(x + 14)
  22. 22.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 1)(x + 28) (c) (x + 1)(x + 10) (d) (x + 4)(x + 7)
  23. 23.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2209 (b) 2809 (c) 2219 (d) 2491
  24. 24.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 576 (b) 266 (c) 384 (d) 256
  25. 25.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 7921 (b) 8101 (c) 8291 (d) 8281
  26. 26.What is the product of the roots of x² − 11x + 30 = 0? (a) 31 (b) 29 (c) 30 (d) 11
  27. 27.If a + b = 20 and ab = 99, what is a² + b²? (a) 301 (b) 400 (c) 202 (d) 598
  28. 28.What are the roots of x² − 9x + 18 = 0? (a) 1 and 8 (b) 1 and 18 (c) 3 and 6 (d) 2 and 9
  29. 29.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4916 (b) 4356 (c) 5486 (d) 5476
  30. 30.Which is the correct factorisation of x² + 11x + 24? (a) (x + 3)(x + 8) (b) (x + 1)(x + 10) (c) (x + 1)(x + 24) (d) (x − 3)(x − 8)
  31. 31.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 271 (c) 1 (d) 361
  32. 32.If a + b = 14 and ab = 40, what is (a − b)²? (a) 196 (b) 36 (c) 6 (d) 116
  33. 33.If a + b = 15 and ab = 36, what is (a − b)²? (a) 81 (b) 225 (c) 9 (d) 153
  34. 34.What is the sum of the roots of x² − 13x + 42 = 0? (a) 13 (b) 42 (c) 14 (d) 12
  35. 35.What is the sum of the roots of x² − 13x + 40 = 0? (a) 40 (b) 14 (c) 12 (d) 13
  36. 36.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 7225 (b) 9035 (c) 9025 (d) 8125
  37. 37.If a + b = 14 and ab = 40, what is a² + b²? (a) 196 (b) 276 (c) 156 (d) 116
  38. 38.If a + b = 19 and ab = 84, what is (a − b)²? (a) 361 (b) 5 (c) 193 (d) 25
  39. 39.What is the sum of the roots of x² − 5x + 6 = 0? (a) 6 (b) 4 (c) 7 (d) 5
  40. 40.If a + b = 15 and ab = 54, what is a² + b²? (a) 225 (b) 117 (c) 333 (d) 171

Answer key

  1. 16
  2. 10
  3. 841
  4. 289
  5. 64
  6. (x + 3)(x + 4)
  7. 961
  8. 48
  9. 169
  10. 181
  11. (x + 8)(x + 9)
  12. 164
  13. 1
  14. 125
  15. 5 and 6
  16. 18
  17. 63
  18. 49
  19. 441
  20. 2 and 5
  21. (x + 6)(x + 9)
  22. (x + 4)(x + 7)
  23. 2209
  24. 256
  25. 8281
  26. 30
  27. 202
  28. 3 and 6
  29. 5476
  30. (x + 3)(x + 8)
  31. 1
  32. 36
  33. 81
  34. 13
  35. 13
  36. 9025
  37. 116
  38. 25
  39. 5
  40. 117

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