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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 94 · 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 + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 1)(x + 15) (c) (x + 1)(x + 7) (d) (x + 3)(x + 5)
  2. 2.If a + b = 12 and ab = 11, what is a² + b²? (a) 144 (b) 133 (c) 166 (d) 122
  3. 3.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 5339 (b) 4909 (c) 4489 (d) 5329
  4. 4.If a + b = 18 and ab = 77, what is (a − b)²? (a) 16 (b) 324 (c) 170 (d) 4
  5. 5.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 375 (b) 235 (c) 625 (d) 225
  6. 6.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9996 (b) 9614 (c) 9604 (d) 10404
  7. 7.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 15 (c) 56 (d) 55
  8. 8.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 1)(x + 12) (c) (x − 6)(x − 7) (d) (x + 6)(x + 7)
  9. 9.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 11025 (b) 9975 (c) 9035 (d) 9025
  10. 10.If a + b = 18 and ab = 80, what is a² + b²? (a) 164 (b) 484 (c) 324 (d) 244
  11. 11.What is the sum of the roots of x² − 9x + 18 = 0? (a) 10 (b) 18 (c) 8 (d) 9
  12. 12.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 18) (b) (x − 2)(x − 9) (c) (x + 2)(x + 9) (d) (x + 1)(x + 10)
  13. 13.If a + b = 13 and ab = 36, what is a² + b²? (a) 133 (b) 241 (c) 169 (d) 97
  14. 14.What is the sum of the roots of x² − 6x + 8 = 0? (a) 5 (b) 7 (c) 6 (d) 8
  15. 15.What are the roots of x² − 5x + 6 = 0? (a) 2 and 4 (b) 2 and 3 (c) 1 and 6 (d) 1 and 4
  16. 16.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 7396 (b) 8836 (c) 8084 (d) 7406
  17. 17.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1774 (b) 1604 (c) 1764 (d) 1444
  18. 18.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2501 (b) 2401 (c) 2611 (d) 2601
  19. 19.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 676 (b) 916 (c) 1166 (d) 1156
  20. 20.Which is the correct factorisation of x² + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 2)(x + 8) (c) (x + 1)(x + 16) (d) (x + 1)(x + 9)
  21. 21.Which is the correct factorisation of x² + 11x + 24? (a) (x + 3)(x + 8) (b) (x − 3)(x − 8) (c) (x + 1)(x + 24) (d) (x + 1)(x + 10)
  22. 22.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7754 (b) 8096 (c) 8464 (d) 7744
  23. 23.If a + b = 11 and ab = 30, what is a² + b²? (a) 121 (b) 91 (c) 61 (d) 181
  24. 24.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 5625 (b) 4875 (c) 4225 (d) 4235
  25. 25.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 3025 (b) 2025 (c) 3035 (d) 2525
  26. 26.If a + b = 7 and ab = 10, what is a² + b²? (a) 29 (b) 49 (c) 69 (d) 39
  27. 27.If a + b = 15 and ab = 56, what is a² + b²? (a) 337 (b) 169 (c) 225 (d) 113
  28. 28.If a + b = 4 and ab = 3, what is (a − b)²? (a) 4 (b) 16 (c) 10 (d) 2
  29. 29.If a + b = 13 and ab = 30, what is (a − b)²? (a) 109 (b) 169 (c) 7 (d) 49
  30. 30.What is the sum of the roots of x² − 14x + 45 = 0? (a) 15 (b) 14 (c) 45 (d) 13
  31. 31.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 71 (c) 17 (d) 72
  32. 32.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2116 (b) 2916 (c) 2926 (d) 2516
  33. 33.If a + b = 20 and ab = 99, what is (a − b)²? (a) 4 (b) 400 (c) 2 (d) 202
  34. 34.If a + b = 3 and ab = 2, what is a² + b²? (a) 13 (b) 7 (c) 9 (d) 5
  35. 35.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 16 (c) 15 (d) 14
  36. 36.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 11035 (b) 9025 (c) 11025 (d) 10025
  37. 37.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 441 (b) 361 (c) 401 (d) 451
  38. 38.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 729 (b) 909 (c) 1099 (d) 1089
  39. 39.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4489 (b) 4891 (c) 4499 (d) 5329
  40. 40.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 72)

Answer key

  1. (x + 3)(x + 5)
  2. 122
  3. 5329
  4. 16
  5. 225
  6. 9604
  7. 56
  8. (x + 6)(x + 7)
  9. 9025
  10. 164
  11. 9
  12. (x + 2)(x + 9)
  13. 97
  14. 6
  15. 2 and 3
  16. 7396
  17. 1764
  18. 2601
  19. 1156
  20. (x + 2)(x + 8)
  21. (x + 3)(x + 8)
  22. 7744
  23. 61
  24. 4225
  25. 3025
  26. 29
  27. 113
  28. 4
  29. 49
  30. 14
  31. 72
  32. 2916
  33. 4
  34. 5
  35. 15
  36. 11025
  37. 441
  38. 1089
  39. 4489
  40. (x + 8)(x + 9)

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