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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 149 · 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.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2819 (b) 2209 (c) 2809 (d) 2509
  2. 2.If a + b = 6 and ab = 8, what is a² + b²? (a) 28 (b) 52 (c) 36 (d) 20
  3. 3.If a + b = 11 and ab = 28, what is a² + b²? (a) 121 (b) 93 (c) 65 (d) 177
  4. 4.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 14 (c) 13 (d) 15
  5. 5.What is the sum of the roots of x² − 11x + 30 = 0? (a) 11 (b) 30 (c) 10 (d) 12
  6. 6.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 896 (b) 794 (c) 784 (d) 1024
  7. 7.If a + b = 21 and ab = 108, what is a² + b²? (a) 225 (b) 657 (c) 333 (d) 441
  8. 8.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 361 (b) 401 (c) 451 (d) 441
  9. 9.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 1 and 16 (c) 1 and 72 (d) 2 and 36
  10. 10.Which is the correct factorisation of x² + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 1)(x + 9) (c) (x + 2)(x + 8) (d) (x + 1)(x + 16)
  11. 11.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3609 (b) 3979 (c) 3969 (d) 3249
  12. 12.If a + b = 18 and ab = 72, what is a² + b²? (a) 252 (b) 180 (c) 468 (d) 324
  13. 13.What is the product of the roots of x² − 6x + 8 = 0? (a) 9 (b) 7 (c) 6 (d) 8
  14. 14.What is the sum of the roots of x² − 10x + 21 = 0? (a) 9 (b) 10 (c) 21 (d) 11
  15. 15.What is the product of the roots of x² − 12x + 27 = 0? (a) 12 (b) 28 (c) 26 (d) 27
  16. 16.What is the product of the roots of x² − 9x + 20 = 0? (a) 21 (b) 20 (c) 19 (d) 9
  17. 17.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x + 7)(x + 8) (d) (x − 7)(x − 8)
  18. 18.What is the product of the roots of x² − 11x + 30 = 0? (a) 31 (b) 30 (c) 29 (d) 11
  19. 19.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 15) (c) (x + 7)(x + 9) (d) (x + 1)(x + 63)
  20. 20.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3854 (b) 3844 (c) 3604 (d) 3364
  21. 21.If a + b = 12 and ab = 20, what is a² + b²? (a) 184 (b) 104 (c) 124 (d) 144
  22. 22.If a + b = 7 and ab = 12, what is (a − b)²? (a) 49 (b) 37 (c) 25 (d) 1
  23. 23.If a + b = 9 and ab = 14, what is (a − b)²? (a) 53 (b) 5 (c) 25 (d) 81
  24. 24.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1616 (b) 1296 (c) 1946 (d) 1936
  25. 25.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8846 (b) 7396 (c) 8116 (d) 8836
  26. 26.What is the product of the roots of x² − 5x + 6 = 0? (a) 7 (b) 6 (c) 8 (d) 5
  27. 27.What are the roots of x² − 14x + 45 = 0? (a) 1 and 45 (b) 1 and 13 (c) 5 and 9 (d) 3 and 15
  28. 28.What is the sum of the roots of x² − 13x + 40 = 0? (a) 14 (b) 13 (c) 40 (d) 12
  29. 29.If a + b = 14 and ab = 48, what is (a − b)²? (a) 100 (b) 4 (c) 2 (d) 196
  30. 30.What are the roots of x² − 13x + 40 = 0? (a) 1 and 12 (b) 2 and 20 (c) 1 and 40 (d) 5 and 8
  31. 31.If a + b = 9 and ab = 20, what is a² + b²? (a) 41 (b) 81 (c) 121 (d) 61
  32. 32.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)
  33. 33.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 289 (b) 409 (c) 539 (d) 529
  34. 34.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 7406 (b) 7396 (c) 8836 (d) 8084
  35. 35.If a + b = 14 and ab = 48, what is a² + b²? (a) 148 (b) 292 (c) 196 (d) 100
  36. 36.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10201 (b) 10001 (c) 9801 (d) 10211
  37. 37.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9991 (b) 9419 (c) 10609 (d) 9409
  38. 38.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 576 (b) 384 (c) 266 (d) 256
  39. 39.If a + b = 15 and ab = 56, what is a² + b²? (a) 113 (b) 225 (c) 337 (d) 169
  40. 40.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 11) (b) (x − 4)(x − 8) (c) (x + 4)(x + 8) (d) (x + 1)(x + 32)

Answer key

  1. 2809
  2. 20
  3. 65
  4. 14
  5. 11
  6. 784
  7. 225
  8. 441
  9. 8 and 9
  10. (x + 2)(x + 8)
  11. 3969
  12. 180
  13. 8
  14. 10
  15. 27
  16. 20
  17. (x + 7)(x + 8)
  18. 30
  19. (x + 7)(x + 9)
  20. 3844
  21. 104
  22. 1
  23. 25
  24. 1936
  25. 8836
  26. 6
  27. 5 and 9
  28. 13
  29. 4
  30. 5 and 8
  31. 41
  32. (x + 8)(x + 9)
  33. 529
  34. 7396
  35. 100
  36. 10201
  37. 9409
  38. 256
  39. 113
  40. (x + 4)(x + 8)

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