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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 806 · 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 = 15 and ab = 54, what is a² + b²? (a) 171 (b) 333 (c) 225 (d) 117
  2. 2.If a + b = 13 and ab = 42, what is a² + b²? (a) 85 (b) 127 (c) 169 (d) 253
  3. 3.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 56) (c) (x + 7)(x + 8) (d) (x + 1)(x + 14)
  4. 4.What is the product of the roots of x² − 15x + 54 = 0? (a) 55 (b) 53 (c) 15 (d) 54
  5. 5.What is the sum of the roots of x² − 8x + 12 = 0? (a) 12 (b) 9 (c) 7 (d) 8
  6. 6.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8116 (b) 8846 (c) 8836 (d) 7396
  7. 7.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10016 (b) 10826 (c) 10816 (d) 9216
  8. 8.What are the roots of x² − 15x + 56 = 0? (a) 2 and 28 (b) 1 and 56 (c) 7 and 8 (d) 1 and 14
  9. 9.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x − 2)(x − 7) (c) (x + 2)(x + 7) (d) (x + 1)(x + 8)
  10. 10.What are the roots of x² − 11x + 30 = 0? (a) 1 and 30 (b) 1 and 10 (c) 2 and 15 (d) 5 and 6
  11. 11.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x − 3)(x − 5) (c) (x + 1)(x + 7) (d) (x + 3)(x + 5)
  12. 12.What is the product of the roots of x² − 9x + 20 = 0? (a) 20 (b) 21 (c) 9 (d) 19
  13. 13.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 896 (b) 784 (c) 794 (d) 1024
  14. 14.What are the roots of x² − 9x + 18 = 0? (a) 2 and 9 (b) 1 and 8 (c) 3 and 6 (d) 1 and 18
  15. 15.If a + b = 18 and ab = 80, what is (a − b)²? (a) 164 (b) 4 (c) 2 (d) 324
  16. 16.If a + b = 9 and ab = 14, what is a² + b²? (a) 109 (b) 81 (c) 67 (d) 53
  17. 17.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6084 (b) 6724 (c) 6404 (d) 6734
  18. 18.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 72 (c) 17 (d) 16
  19. 19.What is the product of the roots of x² − 10x + 16 = 0? (a) 17 (b) 10 (c) 16 (d) 15
  20. 20.If a + b = 8 and ab = 15, what is a² + b²? (a) 64 (b) 34 (c) 94 (d) 49
  21. 21.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6571 (b) 6561 (c) 6401 (d) 6241
  22. 22.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10004 (b) 9604 (c) 10404 (d) 10414
  23. 23.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4235 (b) 4875 (c) 5625 (d) 4225
  24. 24.If a + b = 17 and ab = 70, what is a² + b²? (a) 289 (b) 219 (c) 429 (d) 149
  25. 25.If a + b = 10 and ab = 9, what is (a − b)²? (a) 82 (b) 100 (c) 8 (d) 64
  26. 26.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 5041 (b) 4899 (c) 4771 (d) 4761
  27. 27.What are the roots of x² − 10x + 16 = 0? (a) 1 and 9 (b) 3 and 7 (c) 2 and 8 (d) 1 and 16
  28. 28.What is the sum of the roots of x² − 6x + 8 = 0? (a) 5 (b) 8 (c) 6 (d) 7
  29. 29.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 841 (b) 971 (c) 901 (d) 961
  30. 30.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 409 (b) 289 (c) 529 (d) 539
  31. 31.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 15 (c) 56 (d) 57
  32. 32.If a + b = 13 and ab = 36, what is (a − b)²? (a) 5 (b) 169 (c) 25 (d) 97
  33. 33.What is the product of the roots of x² − 9x + 14 = 0? (a) 13 (b) 14 (c) 9 (d) 15
  34. 34.What is the sum of the roots of x² − 14x + 48 = 0? (a) 14 (b) 13 (c) 48 (d) 15
  35. 35.If a + b = 17 and ab = 70, what is (a − b)²? (a) 149 (b) 3 (c) 9 (d) 289
  36. 36.If a + b = 17 and ab = 72, what is a² + b²? (a) 217 (b) 289 (c) 433 (d) 145
  37. 37.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x + 1)(x + 13) (c) (x + 6)(x + 8) (d) (x − 6)(x − 8)
  38. 38.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
  39. 39.What are the roots of x² − 14x + 48 = 0? (a) 1 and 48 (b) 1 and 13 (c) 2 and 24 (d) 6 and 8
  40. 40.If a + b = 10 and ab = 16, what is (a − b)²? (a) 68 (b) 6 (c) 100 (d) 36

Answer key

  1. 117
  2. 85
  3. (x + 7)(x + 8)
  4. 54
  5. 8
  6. 8836
  7. 10816
  8. 7 and 8
  9. (x + 2)(x + 7)
  10. 5 and 6
  11. (x + 3)(x + 5)
  12. 20
  13. 784
  14. 3 and 6
  15. 4
  16. 53
  17. 6724
  18. 17
  19. 16
  20. 34
  21. 6561
  22. 10404
  23. 4225
  24. 149
  25. 64
  26. 4761
  27. 2 and 8
  28. 6
  29. 961
  30. 529
  31. 56
  32. 25
  33. 14
  34. 14
  35. 9
  36. 145
  37. (x + 6)(x + 8)
  38. (x + 8)(x + 9)
  39. 6 and 8
  40. 36

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