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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 830 · 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 = 8 and ab = 15, what is a² + b²? (a) 94 (b) 64 (c) 49 (d) 34
  2. 2.What are the roots of x² − 12x + 32 = 0? (a) 1 and 11 (b) 4 and 8 (c) 1 and 32 (d) 2 and 16
  3. 3.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x + 1)(x + 16)
  4. 4.If a + b = 6 and ab = 8, what is (a − b)²? (a) 2 (b) 4 (c) 36 (d) 20
  5. 5.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9996 (b) 10404 (c) 9604 (d) 9614
  6. 6.What are the roots of x² − 15x + 56 = 0? (a) 7 and 8 (b) 2 and 28 (c) 1 and 14 (d) 1 and 56
  7. 7.What is the sum of the roots of x² − 7x + 12 = 0? (a) 7 (b) 6 (c) 12 (d) 8
  8. 8.If a + b = 10 and ab = 16, what is a² + b²? (a) 100 (b) 68 (c) 132 (d) 84
  9. 9.If a + b = 13 and ab = 36, what is (a − b)²? (a) 25 (b) 97 (c) 169 (d) 5
  10. 10.If a + b = 17 and ab = 70, what is a² + b²? (a) 289 (b) 429 (c) 219 (d) 149
  11. 11.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x − 7)(x − 8) (c) (x + 1)(x + 56) (d) (x + 7)(x + 8)
  12. 12.If a + b = 9 and ab = 14, what is (a − b)²? (a) 81 (b) 53 (c) 25 (d) 5
  13. 13.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 1)(x + 8) (c) (x + 1)(x + 5) (d) (x + 2)(x + 4)
  14. 14.If a + b = 13 and ab = 12, what is (a − b)²? (a) 121 (b) 145 (c) 169 (d) 11
  15. 15.What is the sum of the roots of x² − 7x + 10 = 0? (a) 10 (b) 8 (c) 7 (d) 6
  16. 16.If a + b = 17 and ab = 72, what is a² + b²? (a) 289 (b) 145 (c) 433 (d) 217
  17. 17.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1296 (b) 1584 (c) 1306 (d) 1936
  18. 18.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x − 2)(x − 6) (c) (x + 2)(x + 6) (d) (x + 1)(x + 7)
  19. 19.If a + b = 16 and ab = 63, what is (a − b)²? (a) 130 (b) 256 (c) 2 (d) 4
  20. 20.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x − 3)(x − 8) (c) (x + 3)(x + 8) (d) (x + 1)(x + 10)
  21. 21.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 5929 (b) 6409 (c) 6889 (d) 6899
  22. 22.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6561 (b) 6399 (c) 6241 (d) 6251
  23. 23.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x − 3)(x − 7) (c) (x + 3)(x + 7) (d) (x + 1)(x + 21)
  24. 24.If a + b = 15 and ab = 56, what is (a − b)²? (a) 225 (b) 169 (c) 113 (d) 1
  25. 25.If a + b = 20 and ab = 96, what is (a − b)²? (a) 208 (b) 400 (c) 4 (d) 16
  26. 26.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 71 (c) 73 (d) 17
  27. 27.What are the roots of x² − 10x + 16 = 0? (a) 1 and 9 (b) 2 and 8 (c) 1 and 16 (d) 3 and 7
  28. 28.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 17 (c) 72 (d) 18
  29. 29.If a + b = 18 and ab = 72, what is (a − b)²? (a) 6 (b) 36 (c) 324 (d) 180
  30. 30.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 225 (b) 625 (c) 375 (d) 235
  31. 31.If a + b = 10 and ab = 9, what is (a − b)²? (a) 100 (b) 8 (c) 64 (d) 82
  32. 32.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1444 (b) 1596 (c) 1454 (d) 1764
  33. 33.If a + b = 17 and ab = 72, what is (a − b)²? (a) 145 (b) 289 (c) 1 (d) 217
  34. 34.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9409 (b) 10609 (c) 9419 (d) 9991
  35. 35.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 5)(x + 6) (c) (x − 5)(x − 6) (d) (x + 1)(x + 10)
  36. 36.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 266 (b) 384 (c) 576 (d) 256
  37. 37.If a + b = 4 and ab = 3, what is (a − b)²? (a) 16 (b) 4 (c) 10 (d) 2
  38. 38.What is the sum of the roots of x² − 10x + 24 = 0? (a) 11 (b) 9 (c) 24 (d) 10
  39. 39.If a + b = 8 and ab = 7, what is a² + b²? (a) 64 (b) 78 (c) 57 (d) 50
  40. 40.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 56 (c) 15 (d) 57

Answer key

  1. 34
  2. 4 and 8
  3. (x + 8)(x + 9)
  4. 4
  5. 9604
  6. 7 and 8
  7. 7
  8. 68
  9. 25
  10. 149
  11. (x + 7)(x + 8)
  12. 25
  13. (x + 2)(x + 4)
  14. 121
  15. 7
  16. 145
  17. 1296
  18. (x + 2)(x + 6)
  19. 4
  20. (x + 3)(x + 8)
  21. 6889
  22. 6241
  23. (x + 3)(x + 7)
  24. 1
  25. 16
  26. 72
  27. 2 and 8
  28. 17
  29. 36
  30. 225
  31. 64
  32. 1444
  33. 1
  34. 9409
  35. (x + 5)(x + 6)
  36. 256
  37. 4
  38. 10
  39. 50
  40. 56

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