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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 990 · 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 75²? (a) 6375 (b) 5625 (c) 7225 (d) 5635
  2. 2.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3609 (b) 3979 (c) 3249 (d) 3969
  3. 3.If a + b = 10 and ab = 21, what is a² + b²? (a) 58 (b) 79 (c) 142 (d) 100
  4. 4.If a + b = 7 and ab = 12, what is a² + b²? (a) 73 (b) 49 (c) 37 (d) 25
  5. 5.If a + b = 13 and ab = 40, what is (a − b)²? (a) 169 (b) 9 (c) 3 (d) 89
  6. 6.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6094 (b) 6396 (c) 6724 (d) 6084
  7. 7.If a + b = 15 and ab = 44, what is (a − b)²? (a) 49 (b) 7 (c) 137 (d) 225
  8. 8.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 8)(x + 9) (c) (x + 1)(x + 16) (d) (x − 8)(x − 9)
  9. 9.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 324 (b) 494 (c) 404 (d) 484
  10. 10.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 529 (b) 289 (c) 409 (d) 539
  11. 11.If a + b = 12 and ab = 35, what is (a − b)²? (a) 74 (b) 144 (c) 4 (d) 2
  12. 12.Which is the correct factorisation of x² + 12x + 32? (a) (x + 4)(x + 8) (b) (x + 1)(x + 11) (c) (x − 4)(x − 8) (d) (x + 1)(x + 32)
  13. 13.If a + b = 9 and ab = 20, what is a² + b²? (a) 121 (b) 61 (c) 41 (d) 81
  14. 14.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 73 (c) 71 (d) 17
  15. 15.What is the sum of the roots of x² − 9x + 20 = 0? (a) 10 (b) 20 (c) 9 (d) 8
  16. 16.What is the product of the roots of x² − 11x + 18 = 0? (a) 19 (b) 11 (c) 17 (d) 18
  17. 17.If a + b = 7 and ab = 10, what is a² + b²? (a) 29 (b) 69 (c) 49 (d) 39
  18. 18.If a + b = 12 and ab = 11, what is (a − b)²? (a) 144 (b) 10 (c) 122 (d) 100
  19. 19.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1531 (b) 1521 (c) 1599 (d) 1681
  20. 20.If a + b = 15 and ab = 36, what is (a − b)²? (a) 81 (b) 153 (c) 225 (d) 9
  21. 21.Which is the correct factorisation of x² + 5x + 6? (a) (x + 2)(x + 3) (b) (x + 1)(x + 4) (c) (x + 1)(x + 6) (d) (x − 2)(x − 3)
  22. 22.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 4106 (b) 3616 (c) 3136 (d) 4096
  23. 23.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x + 1)(x + 10) (c) (x − 5)(x − 6) (d) (x + 1)(x + 30)
  24. 24.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x + 1)(x + 8) (c) (x + 3)(x + 6) (d) (x − 3)(x − 6)
  25. 25.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 1)(x + 8) (c) (x + 1)(x + 14) (d) (x + 2)(x + 7)
  26. 26.What is the sum of the roots of x² − 13x + 36 = 0? (a) 14 (b) 13 (c) 36 (d) 12
  27. 27.What are the roots of x² − 11x + 18 = 0? (a) 1 and 18 (b) 1 and 10 (c) 2 and 9 (d) 3 and 6
  28. 28.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 17 (c) 15 (d) 16
  29. 29.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6899 (b) 6889 (c) 6409 (d) 5929
  30. 30.What are the roots of x² − 8x + 15 = 0? (a) 1 and 15 (b) 2 and 6 (c) 1 and 7 (d) 3 and 5
  31. 31.If a + b = 4 and ab = 3, what is a² + b²? (a) 10 (b) 16 (c) 22 (d) 13
  32. 32.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9975 (b) 9025 (c) 11025 (d) 9035
  33. 33.If a + b = 15 and ab = 50, what is (a − b)²? (a) 225 (b) 125 (c) 25 (d) 5
  34. 34.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1235 (b) 925 (c) 625 (d) 1225
  35. 35.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x + 1)(x + 14) (c) (x + 1)(x + 54) (d) (x − 6)(x − 9)
  36. 36.What is the sum of the roots of x² − 13x + 42 = 0? (a) 14 (b) 12 (c) 13 (d) 42
  37. 37.What is the sum of the roots of x² − 10x + 16 = 0? (a) 16 (b) 9 (c) 11 (d) 10
  38. 38.What are the roots of x² − 7x + 12 = 0? (a) 1 and 12 (b) 2 and 6 (c) 3 and 4 (d) 1 and 6
  39. 39.If a + b = 15 and ab = 56, what is a² + b²? (a) 225 (b) 337 (c) 113 (d) 169
  40. 40.What are the roots of x² − 13x + 40 = 0? (a) 5 and 8 (b) 2 and 20 (c) 1 and 40 (d) 1 and 12

Answer key

  1. 5625
  2. 3969
  3. 58
  4. 25
  5. 9
  6. 6084
  7. 49
  8. (x + 8)(x + 9)
  9. 484
  10. 529
  11. 4
  12. (x + 4)(x + 8)
  13. 41
  14. 72
  15. 9
  16. 18
  17. 29
  18. 100
  19. 1521
  20. 81
  21. (x + 2)(x + 3)
  22. 4096
  23. (x + 5)(x + 6)
  24. (x + 3)(x + 6)
  25. (x + 2)(x + 7)
  26. 13
  27. 2 and 9
  28. 16
  29. 6889
  30. 3 and 5
  31. 10
  32. 9025
  33. 25
  34. 1225
  35. (x + 6)(x + 9)
  36. 13
  37. 10
  38. 3 and 4
  39. 113
  40. 5 and 8

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