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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 824 · 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 = 13 and ab = 40, what is (a − b)²? (a) 89 (b) 169 (c) 9 (d) 3
  2. 2.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3035 (b) 3575 (c) 4225 (d) 3025
  3. 3.If a + b = 15 and ab = 50, what is a² + b²? (a) 175 (b) 225 (c) 125 (d) 325
  4. 4.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 6384 (b) 5776 (c) 5786 (d) 7056
  5. 5.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2509 (b) 2819 (c) 2809 (d) 2209
  6. 6.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2219 (b) 2809 (c) 2491 (d) 2209
  7. 7.If a + b = 10 and ab = 9, what is a² + b²? (a) 82 (b) 118 (c) 100 (d) 91
  8. 8.If a + b = 5 and ab = 6, what is a² + b²? (a) 25 (b) 13 (c) 37 (d) 19
  9. 9.If a + b = 18 and ab = 77, what is (a − b)²? (a) 16 (b) 324 (c) 4 (d) 170
  10. 10.What is the product of the roots of x² − 7x + 12 = 0? (a) 11 (b) 12 (c) 13 (d) 7
  11. 11.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10016 (b) 9216 (c) 10826 (d) 10816
  12. 12.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4366 (b) 4356 (c) 5476 (d) 4884
  13. 13.If a + b = 15 and ab = 36, what is (a − b)²? (a) 225 (b) 81 (c) 153 (d) 9
  14. 14.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1601 (b) 1681 (c) 1691 (d) 1521
  15. 15.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 729 (b) 739 (c) 1089 (d) 891
  16. 16.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
  17. 17.If a + b = 12 and ab = 35, what is (a − b)²? (a) 2 (b) 4 (c) 74 (d) 144
  18. 18.If a + b = 14 and ab = 40, what is a² + b²? (a) 116 (b) 276 (c) 196 (d) 156
  19. 19.What is the sum of the roots of x² − 11x + 30 = 0? (a) 30 (b) 11 (c) 10 (d) 12
  20. 20.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 2)(x + 4) (c) (x − 2)(x − 4) (d) (x + 1)(x + 5)
  21. 21.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 400 (c) 4 (d) 202
  22. 22.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 9 (c) 225 (d) 3
  23. 23.If a + b = 18 and ab = 80, what is a² + b²? (a) 324 (b) 244 (c) 484 (d) 164
  24. 24.If a + b = 13 and ab = 42, what is a² + b²? (a) 85 (b) 253 (c) 127 (d) 169
  25. 25.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 63) (c) (x + 1)(x + 15) (d) (x + 7)(x + 9)
  26. 26.What is the sum of the roots of x² − 8x + 12 = 0? (a) 8 (b) 9 (c) 12 (d) 7
  27. 27.If a + b = 19 and ab = 88, what is (a − b)²? (a) 9 (b) 361 (c) 185 (d) 3
  28. 28.If a + b = 9 and ab = 14, what is (a − b)²? (a) 5 (b) 53 (c) 25 (d) 81
  29. 29.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2401 (b) 2499 (c) 2601 (d) 2411
  30. 30.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 72 (c) 16 (d) 17
  31. 31.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 27) (b) (x − 3)(x − 9) (c) (x + 1)(x + 11) (d) (x + 3)(x + 9)
  32. 32.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 1)(x + 30) (c) (x − 5)(x − 6) (d) (x + 5)(x + 6)
  33. 33.If a + b = 9 and ab = 20, what is (a − b)²? (a) 1 (b) 41 (c) 61 (d) 81
  34. 34.What are the roots of x² − 10x + 21 = 0? (a) 1 and 21 (b) 3 and 7 (c) 2 and 8 (d) 1 and 9
  35. 35.If a + b = 19 and ab = 88, what is a² + b²? (a) 361 (b) 537 (c) 185 (d) 273
  36. 36.If a + b = 5 and ab = 4, what is a² + b²? (a) 25 (b) 33 (c) 17 (d) 21
  37. 37.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1225 (b) 2025 (c) 1575 (d) 1235
  38. 38.If a + b = 15 and ab = 56, what is a² + b²? (a) 225 (b) 113 (c) 169 (d) 337
  39. 39.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x − 6)(x − 8) (c) (x + 6)(x + 8) (d) (x + 1)(x + 13)
  40. 40.If a + b = 12 and ab = 35, what is a² + b²? (a) 109 (b) 214 (c) 74 (d) 144

Answer key

  1. 9
  2. 3025
  3. 125
  4. 5776
  5. 2809
  6. 2209
  7. 82
  8. 13
  9. 16
  10. 12
  11. 10816
  12. 4356
  13. 81
  14. 1681
  15. 729
  16. (x + 8)(x + 9)
  17. 4
  18. 116
  19. 11
  20. (x + 2)(x + 4)
  21. 4
  22. 9
  23. 164
  24. 85
  25. (x + 7)(x + 9)
  26. 8
  27. 9
  28. 25
  29. 2401
  30. 17
  31. (x + 3)(x + 9)
  32. (x + 5)(x + 6)
  33. 1
  34. 3 and 7
  35. 185
  36. 17
  37. 1225
  38. 113
  39. (x + 6)(x + 8)
  40. 74

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