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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 701 · 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 = 17 and ab = 72, what is a² + b²? (a) 217 (b) 433 (c) 289 (d) 145
  2. 2.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 2 (c) 202 (d) 4
  3. 3.If a + b = 11 and ab = 24, what is a² + b²? (a) 97 (b) 121 (c) 169 (d) 73
  4. 4.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 1)(x + 72) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)
  5. 5.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1764 (b) 1596 (c) 1444 (d) 1454
  6. 6.What is the sum of the roots of x² − 14x + 45 = 0? (a) 45 (b) 13 (c) 15 (d) 14
  7. 7.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x + 1)(x + 10) (c) (x + 2)(x + 5) (d) (x − 2)(x − 5)
  8. 8.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 441 (b) 361 (c) 401 (d) 451
  9. 9.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 529 (b) 409 (c) 289 (d) 539
  10. 10.What are the roots of x² − 11x + 18 = 0? (a) 3 and 6 (b) 1 and 18 (c) 1 and 10 (d) 2 and 9
  11. 11.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 4)(x + 7) (c) (x + 1)(x + 10) (d) (x + 1)(x + 28)
  12. 12.If a + b = 9 and ab = 8, what is (a − b)²? (a) 65 (b) 7 (c) 81 (d) 49
  13. 13.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)
  14. 14.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 73 (c) 72 (d) 71
  15. 15.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 625 (b) 225 (c) 375 (d) 235
  16. 16.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x + 1)(x + 9) (c) (x + 3)(x + 7) (d) (x − 3)(x − 7)
  17. 17.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9035 (b) 9025 (c) 9975 (d) 11025
  18. 18.If a + b = 13 and ab = 12, what is a² + b²? (a) 193 (b) 157 (c) 145 (d) 169
  19. 19.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1089 (b) 909 (c) 729 (d) 1099
  20. 20.If a + b = 17 and ab = 60, what is (a − b)²? (a) 169 (b) 49 (c) 7 (d) 289
  21. 21.What are the roots of x² − 12x + 27 = 0? (a) 1 and 11 (b) 3 and 9 (c) 1 and 27 (d) 2 and 10
  22. 22.If a + b = 16 and ab = 60, what is a² + b²? (a) 136 (b) 376 (c) 256 (d) 196
  23. 23.If a + b = 10 and ab = 24, what is a² + b²? (a) 100 (b) 52 (c) 148 (d) 76
  24. 24.What is the sum of the roots of x² − 7x + 12 = 0? (a) 7 (b) 6 (c) 12 (d) 8
  25. 25.If a + b = 15 and ab = 56, what is (a − b)²? (a) 113 (b) 225 (c) 1 (d) 169
  26. 26.If a + b = 17 and ab = 66, what is (a − b)²? (a) 25 (b) 289 (c) 5 (d) 157
  27. 27.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6409 (b) 6899 (c) 5929 (d) 6889
  28. 28.What is the product of the roots of x² − 14x + 48 = 0? (a) 49 (b) 47 (c) 48 (d) 14
  29. 29.If a + b = 17 and ab = 70, what is (a − b)²? (a) 289 (b) 3 (c) 9 (d) 149
  30. 30.What is the sum of the roots of x² − 12x + 32 = 0? (a) 11 (b) 13 (c) 32 (d) 12
  31. 31.If a + b = 19 and ab = 88, what is a² + b²? (a) 537 (b) 273 (c) 361 (d) 185
  32. 32.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 14 (c) 15 (d) 16
  33. 33.If a + b = 17 and ab = 72, what is (a − b)²? (a) 145 (b) 289 (c) 217 (d) 1
  34. 34.If a + b = 9 and ab = 20, what is a² + b²? (a) 121 (b) 41 (c) 61 (d) 81
  35. 35.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x + 1)(x + 15) (c) (x + 3)(x + 5) (d) (x − 3)(x − 5)
  36. 36.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 72 (c) 16 (d) 17
  37. 37.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 225 (c) 657 (d) 333
  38. 38.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9614 (b) 9996 (c) 9604 (d) 10404
  39. 39.What is the sum of the roots of x² − 13x + 42 = 0? (a) 42 (b) 12 (c) 14 (d) 13
  40. 40.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x − 2)(x − 9) (c) (x + 1)(x + 18) (d) (x + 2)(x + 9)

Answer key

  1. 145
  2. 4
  3. 73
  4. (x + 8)(x + 9)
  5. 1444
  6. 14
  7. (x + 2)(x + 5)
  8. 441
  9. 529
  10. 2 and 9
  11. (x + 4)(x + 7)
  12. 49
  13. (x + 3)(x + 9)
  14. 72
  15. 225
  16. (x + 3)(x + 7)
  17. 9025
  18. 145
  19. 1089
  20. 49
  21. 3 and 9
  22. 136
  23. 52
  24. 7
  25. 1
  26. 25
  27. 6889
  28. 48
  29. 9
  30. 12
  31. 185
  32. 15
  33. 1
  34. 41
  35. (x + 3)(x + 5)
  36. 17
  37. 225
  38. 9604
  39. 13
  40. (x + 2)(x + 9)

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