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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 592 · 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.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 8)(x + 9) (d) (x + 1)(x + 16)
  2. 2.If a + b = 11 and ab = 10, what is a² + b²? (a) 111 (b) 121 (c) 141 (d) 101
  3. 3.If a + b = 14 and ab = 40, what is a² + b²? (a) 196 (b) 116 (c) 276 (d) 156
  4. 4.What are the roots of x² − 11x + 30 = 0? (a) 5 and 6 (b) 1 and 10 (c) 2 and 15 (d) 1 and 30
  5. 5.If a + b = 15 and ab = 50, what is a² + b²? (a) 325 (b) 225 (c) 175 (d) 125
  6. 6.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 484 (b) 324 (c) 404 (d) 494
  7. 7.If a + b = 20 and ab = 99, what is a² + b²? (a) 202 (b) 598 (c) 400 (d) 301
  8. 8.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 5929 (b) 6899 (c) 6409 (d) 6889
  9. 9.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x − 6)(x − 8) (c) (x + 1)(x + 13) (d) (x + 1)(x + 48)
  10. 10.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6724 (b) 6094 (c) 6396 (d) 6084
  11. 11.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 2 and 36 (c) 1 and 16 (d) 8 and 9
  12. 12.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1225 (b) 925 (c) 1235 (d) 625
  13. 13.What is the sum of the roots of x² − 5x + 6 = 0? (a) 7 (b) 6 (c) 5 (d) 4
  14. 14.If a + b = 15 and ab = 44, what is a² + b²? (a) 225 (b) 137 (c) 181 (d) 313
  15. 15.If a + b = 18 and ab = 72, what is a² + b²? (a) 252 (b) 468 (c) 180 (d) 324
  16. 16.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 784 (b) 1034 (c) 904 (d) 1024
  17. 17.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1609 (b) 1859 (c) 1369 (d) 1849
  18. 18.What is the product of the roots of x² − 10x + 24 = 0? (a) 10 (b) 24 (c) 23 (d) 25
  19. 19.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8846 (b) 7396 (c) 8116 (d) 8836
  20. 20.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 9) (b) (x + 1)(x + 24) (c) (x − 4)(x − 6) (d) (x + 4)(x + 6)
  21. 21.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 441 (b) 451 (c) 401 (d) 361
  22. 22.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x + 5)(x + 8) (c) (x + 1)(x + 40) (d) (x − 5)(x − 8)
  23. 23.If a + b = 9 and ab = 18, what is a² + b²? (a) 81 (b) 45 (c) 117 (d) 63
  24. 24.What is the sum of the roots of x² − 10x + 24 = 0? (a) 11 (b) 24 (c) 10 (d) 9
  25. 25.If a + b = 15 and ab = 56, what is (a − b)²? (a) 1 (b) 225 (c) 169 (d) 113
  26. 26.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x + 1)(x + 7) (c) (x + 3)(x + 5) (d) (x − 3)(x − 5)
  27. 27.If a + b = 12 and ab = 32, what is (a − b)²? (a) 16 (b) 144 (c) 4 (d) 80
  28. 28.If a + b = 12 and ab = 32, what is a² + b²? (a) 112 (b) 208 (c) 80 (d) 144
  29. 29.If a + b = 8 and ab = 15, what is a² + b²? (a) 64 (b) 49 (c) 94 (d) 34
  30. 30.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7225 (b) 8075 (c) 7235 (d) 9025
  31. 31.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3481 (b) 3721 (c) 3731 (d) 3601
  32. 32.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3625 (b) 3025 (c) 4235 (d) 4225
  33. 33.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 1)(x + 10) (c) (x + 5)(x + 6) (d) (x − 5)(x − 6)
  34. 34.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 17 (c) 16 (d) 72
  35. 35.If a + b = 13 and ab = 40, what is (a − b)²? (a) 89 (b) 9 (c) 169 (d) 3
  36. 36.If a + b = 9 and ab = 8, what is a² + b²? (a) 97 (b) 81 (c) 65 (d) 73
  37. 37.What is the product of the roots of x² − 13x + 42 = 0? (a) 43 (b) 42 (c) 41 (d) 13
  38. 38.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3136 (b) 4096 (c) 4106 (d) 3616
  39. 39.If a + b = 7 and ab = 6, what is a² + b²? (a) 61 (b) 43 (c) 49 (d) 37
  40. 40.If a + b = 12 and ab = 35, what is (a − b)²? (a) 2 (b) 4 (c) 74 (d) 144

Answer key

  1. (x + 8)(x + 9)
  2. 101
  3. 116
  4. 5 and 6
  5. 125
  6. 484
  7. 202
  8. 6889
  9. (x + 6)(x + 8)
  10. 6084
  11. 8 and 9
  12. 1225
  13. 5
  14. 137
  15. 180
  16. 1024
  17. 1849
  18. 24
  19. 8836
  20. (x + 4)(x + 6)
  21. 441
  22. (x + 5)(x + 8)
  23. 45
  24. 10
  25. 1
  26. (x + 3)(x + 5)
  27. 16
  28. 80
  29. 34
  30. 7225
  31. 3721
  32. 4225
  33. (x + 5)(x + 6)
  34. 17
  35. 9
  36. 65
  37. 42
  38. 4096
  39. 37
  40. 4

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