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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 387 · 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 = 14 and ab = 24, what is a² + b²? (a) 196 (b) 148 (c) 172 (d) 244
  2. 2.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 8 and 9 (c) 1 and 16 (d) 1 and 72
  3. 3.What is the product of the roots of x² − 11x + 30 = 0? (a) 11 (b) 31 (c) 29 (d) 30
  4. 4.If a + b = 6 and ab = 8, what is a² + b²? (a) 28 (b) 52 (c) 36 (d) 20
  5. 5.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 72 (c) 17 (d) 16
  6. 6.What is the product of the roots of x² − 5x + 6 = 0? (a) 5 (b) 6 (c) 8 (d) 7
  7. 7.If a + b = 20 and ab = 99, what is (a − b)²? (a) 4 (b) 202 (c) 400 (d) 2
  8. 8.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7744 (b) 7754 (c) 8096 (d) 8464
  9. 9.What is the sum of the roots of x² − 12x + 27 = 0? (a) 27 (b) 11 (c) 13 (d) 12
  10. 10.If a + b = 9 and ab = 20, what is a² + b²? (a) 81 (b) 61 (c) 121 (d) 41
  11. 11.If a + b = 19 and ab = 88, what is a² + b²? (a) 185 (b) 537 (c) 361 (d) 273
  12. 12.What is the product of the roots of x² − 9x + 18 = 0? (a) 9 (b) 19 (c) 18 (d) 17
  13. 13.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 73 (c) 71 (d) 72
  14. 14.If a + b = 19 and ab = 88, what is (a − b)²? (a) 9 (b) 185 (c) 3 (d) 361
  15. 15.If a + b = 17 and ab = 66, what is (a − b)²? (a) 157 (b) 5 (c) 25 (d) 289
  16. 16.If a + b = 15 and ab = 56, what is a² + b²? (a) 225 (b) 113 (c) 169 (d) 337
  17. 17.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x + 2)(x + 4) (c) (x + 1)(x + 8) (d) (x − 2)(x − 4)
  18. 18.What is the sum of the roots of x² − 8x + 15 = 0? (a) 15 (b) 9 (c) 7 (d) 8
  19. 19.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8649 (b) 8659 (c) 7569 (d) 8109
  20. 20.If a + b = 14 and ab = 48, what is (a − b)²? (a) 100 (b) 196 (c) 2 (d) 4
  21. 21.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)
  22. 22.What are the roots of x² − 16x + 63 = 0? (a) 1 and 63 (b) 7 and 9 (c) 3 and 21 (d) 1 and 15
  23. 23.If a + b = 10 and ab = 9, what is a² + b²? (a) 82 (b) 91 (c) 100 (d) 118
  24. 24.If a + b = 10 and ab = 21, what is (a − b)²? (a) 4 (b) 16 (c) 100 (d) 58
  25. 25.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x + 3)(x + 6) (c) (x − 3)(x − 6) (d) (x + 1)(x + 18)
  26. 26.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 6)(x + 9) (c) (x − 6)(x − 9) (d) (x + 1)(x + 14)
  27. 27.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9025 (b) 9035 (c) 9975 (d) 11025
  28. 28.If a + b = 13 and ab = 12, what is (a − b)²? (a) 145 (b) 121 (c) 169 (d) 11
  29. 29.If a + b = 14 and ab = 45, what is (a − b)²? (a) 196 (b) 106 (c) 16 (d) 4
  30. 30.What is the product of the roots of x² − 9x + 14 = 0? (a) 15 (b) 13 (c) 14 (d) 9
  31. 31.If a + b = 15 and ab = 44, what is a² + b²? (a) 137 (b) 225 (c) 181 (d) 313
  32. 32.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 256 (b) 384 (c) 266 (d) 576
  33. 33.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 8084 (b) 7396 (c) 7406 (d) 8836
  34. 34.Which is the correct factorisation of x² + 11x + 24? (a) (x − 3)(x − 8) (b) (x + 1)(x + 10) (c) (x + 3)(x + 8) (d) (x + 1)(x + 24)
  35. 35.If a + b = 7 and ab = 12, what is (a − b)²? (a) 37 (b) 25 (c) 49 (d) 1
  36. 36.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3035 (b) 3575 (c) 3025 (d) 4225
  37. 37.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 9801 (b) 10201 (c) 10211 (d) 10001
  38. 38.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 56 (c) 15 (d) 14
  39. 39.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 5486 (b) 5476 (c) 4356 (d) 4916
  40. 40.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 625 (b) 235 (c) 225 (d) 375

Answer key

  1. 148
  2. 8 and 9
  3. 30
  4. 20
  5. 17
  6. 6
  7. 4
  8. 7744
  9. 12
  10. 41
  11. 185
  12. 18
  13. 72
  14. 9
  15. 25
  16. 113
  17. (x + 2)(x + 4)
  18. 8
  19. 8649
  20. 4
  21. (x + 6)(x + 8)
  22. 7 and 9
  23. 82
  24. 16
  25. (x + 3)(x + 6)
  26. (x + 6)(x + 9)
  27. 9025
  28. 121
  29. 16
  30. 14
  31. 137
  32. 256
  33. 7396
  34. (x + 3)(x + 8)
  35. 1
  36. 3025
  37. 10201
  38. 15
  39. 5476
  40. 225

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