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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 378 · 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 + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
  2. 2.If a + b = 17 and ab = 72, what is (a − b)²? (a) 289 (b) 145 (c) 217 (d) 1
  3. 3.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x + 1)(x + 48) (c) (x − 6)(x − 8) (d) (x + 1)(x + 13)
  4. 4.What is the sum of the roots of x² − 7x + 10 = 0? (a) 6 (b) 7 (c) 8 (d) 10
  5. 5.If a + b = 21 and ab = 108, what is a² + b²? (a) 225 (b) 441 (c) 657 (d) 333
  6. 6.What is the product of the roots of x² − 7x + 10 = 0? (a) 7 (b) 10 (c) 11 (d) 9
  7. 7.If a + b = 6 and ab = 8, what is a² + b²? (a) 52 (b) 36 (c) 20 (d) 28
  8. 8.What are the roots of x² − 15x + 54 = 0? (a) 1 and 54 (b) 2 and 27 (c) 1 and 14 (d) 6 and 9
  9. 9.If a + b = 7 and ab = 12, what is a² + b²? (a) 37 (b) 25 (c) 49 (d) 73
  10. 10.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3481 (b) 3491 (c) 3599 (d) 3721
  11. 11.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 961 (b) 899 (c) 851 (d) 841
  12. 12.Which is the correct factorisation of x² + 7x + 10? (a) (x + 2)(x + 5) (b) (x − 2)(x − 5) (c) (x + 1)(x + 10) (d) (x + 1)(x + 6)
  13. 13.If a + b = 15 and ab = 50, what is (a − b)²? (a) 225 (b) 5 (c) 125 (d) 25
  14. 14.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x − 7)(x − 8) (c) (x + 1)(x + 56) (d) (x + 7)(x + 8)
  15. 15.What is the product of the roots of x² − 13x + 42 = 0? (a) 41 (b) 43 (c) 42 (d) 13
  16. 16.If a + b = 14 and ab = 48, what is (a − b)²? (a) 4 (b) 100 (c) 196 (d) 2
  17. 17.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 4 (c) 202 (d) 400
  18. 18.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 961 (b) 901 (c) 841 (d) 971
  19. 19.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8649 (b) 7569 (c) 7579 (d) 8091
  20. 20.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 5635 (b) 5625 (c) 4925 (d) 4225
  21. 21.What are the roots of x² − 11x + 28 = 0? (a) 4 and 7 (b) 2 and 14 (c) 1 and 10 (d) 1 and 28
  22. 22.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2611 (b) 2601 (c) 2401 (d) 2501
  23. 23.If a + b = 17 and ab = 70, what is (a − b)²? (a) 9 (b) 149 (c) 289 (d) 3
  24. 24.If a + b = 11 and ab = 28, what is a² + b²? (a) 177 (b) 65 (c) 121 (d) 93
  25. 25.What is the product of the roots of x² − 15x + 54 = 0? (a) 54 (b) 15 (c) 55 (d) 53
  26. 26.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2916 (b) 2126 (c) 2116 (d) 2484
  27. 27.If a + b = 15 and ab = 54, what is a² + b²? (a) 117 (b) 333 (c) 225 (d) 171
  28. 28.If a + b = 10 and ab = 9, what is (a − b)²? (a) 100 (b) 8 (c) 82 (d) 64
  29. 29.If a + b = 15 and ab = 36, what is a² + b²? (a) 297 (b) 189 (c) 153 (d) 225
  30. 30.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10211 (b) 9801 (c) 10201 (d) 10001
  31. 31.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1859 (b) 1849 (c) 1369 (d) 1609
  32. 32.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 2 and 28 (c) 1 and 14 (d) 7 and 8
  33. 33.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6409 (b) 6899 (c) 6889 (d) 5929
  34. 34.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 625 (b) 635 (c) 875 (d) 1225
  35. 35.If a + b = 11 and ab = 18, what is (a − b)²? (a) 7 (b) 121 (c) 85 (d) 49
  36. 36.What are the roots of x² − 12x + 27 = 0? (a) 1 and 27 (b) 2 and 10 (c) 3 and 9 (d) 1 and 11
  37. 37.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8101 (b) 8291 (c) 7921 (d) 8281
  38. 38.If a + b = 18 and ab = 77, what is (a − b)²? (a) 4 (b) 170 (c) 16 (d) 324
  39. 39.If a + b = 18 and ab = 80, what is (a − b)²? (a) 4 (b) 2 (c) 324 (d) 164
  40. 40.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 5184 (b) 4634 (c) 4896 (d) 4624

Answer key

  1. (x + 8)(x + 9)
  2. 1
  3. (x + 6)(x + 8)
  4. 7
  5. 225
  6. 10
  7. 20
  8. 6 and 9
  9. 25
  10. 3481
  11. 841
  12. (x + 2)(x + 5)
  13. 25
  14. (x + 7)(x + 8)
  15. 42
  16. 4
  17. 4
  18. 961
  19. 7569
  20. 5625
  21. 4 and 7
  22. 2601
  23. 9
  24. 65
  25. 54
  26. 2116
  27. 117
  28. 64
  29. 153
  30. 10201
  31. 1849
  32. 7 and 8
  33. 6889
  34. 625
  35. 49
  36. 3 and 9
  37. 8281
  38. 16
  39. 4
  40. 4624

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