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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 582 · 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 + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 1)(x + 48) (c) (x + 6)(x + 8) (d) (x − 6)(x − 8)
  2. 2.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1454 (b) 1444 (c) 1764 (d) 1596
  3. 3.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)
  4. 4.If a + b = 14 and ab = 45, what is (a − b)²? (a) 106 (b) 16 (c) 4 (d) 196
  5. 5.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 2 and 36 (c) 8 and 9 (d) 1 and 16
  6. 6.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2714 (b) 2504 (c) 2304 (d) 2704
  7. 7.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 11025 (b) 9025 (c) 10025 (d) 11035
  8. 8.If a + b = 13 and ab = 12, what is (a − b)²? (a) 11 (b) 169 (c) 145 (d) 121
  9. 9.If a + b = 6 and ab = 8, what is (a − b)²? (a) 20 (b) 36 (c) 2 (d) 4
  10. 10.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 56 (c) 16 (d) 15
  11. 11.What is the product of the roots of x² − 12x + 32 = 0? (a) 31 (b) 12 (c) 33 (d) 32
  12. 12.What is the sum of the roots of x² − 10x + 21 = 0? (a) 21 (b) 11 (c) 9 (d) 10
  13. 13.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7235 (b) 8075 (c) 9025 (d) 7225
  14. 14.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2314 (b) 2704 (c) 2496 (d) 2304
  15. 15.If a + b = 19 and ab = 88, what is a² + b²? (a) 361 (b) 537 (c) 273 (d) 185
  16. 16.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3136 (b) 3146 (c) 3584 (d) 4096
  17. 17.Which is the correct factorisation of x² + 12x + 32? (a) (x + 4)(x + 8) (b) (x − 4)(x − 8) (c) (x + 1)(x + 32) (d) (x + 1)(x + 11)
  18. 18.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x + 1)(x + 42) (c) (x + 1)(x + 12) (d) (x − 6)(x − 7)
  19. 19.What are the roots of x² − 12x + 27 = 0? (a) 1 and 11 (b) 2 and 10 (c) 1 and 27 (d) 3 and 9
  20. 20.If a + b = 18 and ab = 72, what is (a − b)²? (a) 324 (b) 6 (c) 36 (d) 180
  21. 21.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 7396 (b) 8846 (c) 8116 (d) 8836
  22. 22.If a + b = 10 and ab = 24, what is a² + b²? (a) 100 (b) 52 (c) 76 (d) 148
  23. 23.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 899 (b) 961 (c) 841 (d) 851
  24. 24.If a + b = 7 and ab = 12, what is (a − b)²? (a) 49 (b) 37 (c) 25 (d) 1
  25. 25.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 6)(x + 9) (c) (x + 1)(x + 14) (d) (x − 6)(x − 9)
  26. 26.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 441 (c) 3 (d) 9
  27. 27.What is the product of the roots of x² − 10x + 21 = 0? (a) 10 (b) 20 (c) 21 (d) 22
  28. 28.If a + b = 17 and ab = 66, what is (a − b)²? (a) 5 (b) 25 (c) 289 (d) 157
  29. 29.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 71 (c) 72 (d) 73
  30. 30.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 10609 (b) 9419 (c) 9991 (d) 9409
  31. 31.If a + b = 12 and ab = 32, what is (a − b)²? (a) 4 (b) 80 (c) 16 (d) 144
  32. 32.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8464 (b) 8104 (c) 7744 (d) 8474
  33. 33.If a + b = 13 and ab = 30, what is (a − b)²? (a) 49 (b) 109 (c) 7 (d) 169
  34. 34.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8096 (b) 8464 (c) 7754 (d) 7744
  35. 35.If a + b = 13 and ab = 30, what is a² + b²? (a) 169 (b) 109 (c) 139 (d) 229
  36. 36.Which is the correct factorisation of x² + 12x + 27? (a) (x + 3)(x + 9) (b) (x + 1)(x + 11) (c) (x + 1)(x + 27) (d) (x − 3)(x − 9)
  37. 37.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 7)(x + 8) (c) (x − 7)(x − 8) (d) (x + 1)(x + 56)
  38. 38.What is the product of the roots of x² − 8x + 12 = 0? (a) 11 (b) 12 (c) 13 (d) 8
  39. 39.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 729 (b) 891 (c) 1089 (d) 739
  40. 40.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 17 (c) 15 (d) 16

Answer key

  1. (x + 6)(x + 8)
  2. 1444
  3. (x + 8)(x + 9)
  4. 16
  5. 8 and 9
  6. 2704
  7. 11025
  8. 121
  9. 4
  10. 15
  11. 32
  12. 10
  13. 7225
  14. 2304
  15. 185
  16. 3136
  17. (x + 4)(x + 8)
  18. (x + 6)(x + 7)
  19. 3 and 9
  20. 36
  21. 8836
  22. 52
  23. 841
  24. 1
  25. (x + 6)(x + 9)
  26. 9
  27. 21
  28. 25
  29. 72
  30. 9409
  31. 16
  32. 8464
  33. 49
  34. 7744
  35. 109
  36. (x + 3)(x + 9)
  37. (x + 7)(x + 8)
  38. 12
  39. 729
  40. 16

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