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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 189 · 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.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 73 (c) 71 (d) 72
  2. 2.What is the sum of the roots of x² − 12x + 27 = 0? (a) 27 (b) 11 (c) 13 (d) 12
  3. 3.If a + b = 15 and ab = 54, what is (a − b)²? (a) 9 (b) 225 (c) 3 (d) 117
  4. 4.If a + b = 5 and ab = 6, what is a² + b²? (a) 13 (b) 37 (c) 25 (d) 19
  5. 5.If a + b = 21 and ab = 108, what is (a − b)²? (a) 9 (b) 3 (c) 225 (d) 441
  6. 6.If a + b = 7 and ab = 6, what is a² + b²? (a) 61 (b) 49 (c) 37 (d) 43
  7. 7.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 10025 (b) 9025 (c) 11025 (d) 11035
  8. 8.If a + b = 11 and ab = 28, what is a² + b²? (a) 93 (b) 65 (c) 177 (d) 121
  9. 9.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)
  10. 10.What is the product of the roots of x² − 10x + 16 = 0? (a) 17 (b) 10 (c) 15 (d) 16
  11. 11.If a + b = 20 and ab = 99, what is a² + b²? (a) 202 (b) 598 (c) 400 (d) 301
  12. 12.If a + b = 17 and ab = 72, what is (a − b)²? (a) 145 (b) 1 (c) 217 (d) 289
  13. 13.What is the sum of the roots of x² − 13x + 42 = 0? (a) 14 (b) 42 (c) 13 (d) 12
  14. 14.If a + b = 19 and ab = 88, what is (a − b)²? (a) 3 (b) 361 (c) 185 (d) 9
  15. 15.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1616 (b) 1296 (c) 1946 (d) 1936
  16. 16.What are the roots of x² − 8x + 12 = 0? (a) 3 and 4 (b) 1 and 7 (c) 1 and 12 (d) 2 and 6
  17. 17.What is the sum of the roots of x² − 12x + 35 = 0? (a) 13 (b) 12 (c) 35 (d) 11
  18. 18.If a + b = 18 and ab = 80, what is a² + b²? (a) 244 (b) 324 (c) 484 (d) 164
  19. 19.What is the sum of the roots of x² − 14x + 45 = 0? (a) 15 (b) 13 (c) 14 (d) 45
  20. 20.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x + 1)(x + 21) (c) (x − 3)(x − 7) (d) (x + 3)(x + 7)
  21. 21.If a + b = 10 and ab = 24, what is (a − b)²? (a) 100 (b) 52 (c) 4 (d) 2
  22. 22.If a + b = 16 and ab = 60, what is (a − b)²? (a) 256 (b) 4 (c) 16 (d) 136
  23. 23.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)
  24. 24.What is the product of the roots of x² − 15x + 54 = 0? (a) 55 (b) 54 (c) 15 (d) 53
  25. 25.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6401 (b) 6241 (c) 6561 (d) 6571
  26. 26.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x + 2)(x + 3) (c) (x − 2)(x − 3) (d) (x + 1)(x + 6)
  27. 27.If a + b = 17 and ab = 70, what is a² + b²? (a) 219 (b) 149 (c) 289 (d) 429
  28. 28.If a + b = 17 and ab = 70, what is (a − b)²? (a) 149 (b) 3 (c) 9 (d) 289
  29. 29.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 18 (d) 16
  30. 30.What is the sum of the roots of x² − 13x + 40 = 0? (a) 40 (b) 13 (c) 12 (d) 14
  31. 31.What are the roots of x² − 7x + 12 = 0? (a) 3 and 4 (b) 1 and 12 (c) 1 and 6 (d) 2 and 6
  32. 32.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6724 (b) 6396 (c) 6094 (d) 6084
  33. 33.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 7396 (b) 8116 (c) 8846 (d) 8836
  34. 34.If a + b = 16 and ab = 48, what is a² + b²? (a) 256 (b) 208 (c) 352 (d) 160
  35. 35.What are the roots of x² − 10x + 16 = 0? (a) 3 and 7 (b) 1 and 16 (c) 2 and 8 (d) 1 and 9
  36. 36.What is the product of the roots of x² − 9x + 14 = 0? (a) 15 (b) 9 (c) 14 (d) 13
  37. 37.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2601 (b) 2499 (c) 2411 (d) 2401
  38. 38.If a + b = 20 and ab = 96, what is (a − b)²? (a) 16 (b) 208 (c) 400 (d) 4
  39. 39.What are the roots of x² − 13x + 40 = 0? (a) 2 and 20 (b) 1 and 40 (c) 5 and 8 (d) 1 and 12
  40. 40.What is the sum of the roots of x² − 6x + 8 = 0? (a) 5 (b) 8 (c) 6 (d) 7

Answer key

  1. 72
  2. 12
  3. 9
  4. 13
  5. 9
  6. 37
  7. 11025
  8. 65
  9. (x + 3)(x + 8)
  10. 16
  11. 202
  12. 1
  13. 13
  14. 9
  15. 1936
  16. 2 and 6
  17. 12
  18. 164
  19. 14
  20. (x + 3)(x + 7)
  21. 4
  22. 16
  23. (x + 8)(x + 9)
  24. 54
  25. 6561
  26. (x + 2)(x + 3)
  27. 149
  28. 9
  29. 17
  30. 13
  31. 3 and 4
  32. 6084
  33. 8836
  34. 160
  35. 2 and 8
  36. 14
  37. 2401
  38. 16
  39. 5 and 8
  40. 6

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