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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 544 · 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 + 13x + 42? (a) (x + 1)(x + 42) (b) (x − 6)(x − 7) (c) (x + 6)(x + 7) (d) (x + 1)(x + 12)
  2. 2.What are the roots of x² − 8x + 15 = 0? (a) 2 and 6 (b) 3 and 5 (c) 1 and 7 (d) 1 and 15
  3. 3.What is the sum of the roots of x² − 11x + 18 = 0? (a) 12 (b) 18 (c) 11 (d) 10
  4. 4.Which is the correct factorisation of x² + 9x + 18? (a) (x + 3)(x + 6) (b) (x + 1)(x + 18) (c) (x + 1)(x + 8) (d) (x − 3)(x − 6)
  5. 5.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x + 1)(x + 12) (c) (x − 5)(x − 8) (d) (x + 5)(x + 8)
  6. 6.What are the roots of x² − 13x + 36 = 0? (a) 1 and 12 (b) 1 and 36 (c) 4 and 9 (d) 2 and 18
  7. 7.What is the sum of the roots of x² − 7x + 10 = 0? (a) 10 (b) 8 (c) 7 (d) 6
  8. 8.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3575 (b) 3025 (c) 3035 (d) 4225
  9. 9.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1609 (b) 1849 (c) 1859 (d) 1369
  10. 10.If a + b = 10 and ab = 16, what is a² + b²? (a) 68 (b) 84 (c) 100 (d) 132
  11. 11.What is the sum of the roots of x² − 12x + 27 = 0? (a) 27 (b) 11 (c) 13 (d) 12
  12. 12.Which is the correct factorisation of x² + 11x + 28? (a) (x + 4)(x + 7) (b) (x + 1)(x + 10) (c) (x − 4)(x − 7) (d) (x + 1)(x + 28)
  13. 13.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 3025 (b) 2025 (c) 3035 (d) 2525
  14. 14.What is the sum of the roots of x² − 5x + 6 = 0? (a) 5 (b) 6 (c) 7 (d) 4
  15. 15.What is the product of the roots of x² − 10x + 24 = 0? (a) 25 (b) 24 (c) 23 (d) 10
  16. 16.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)
  17. 17.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x + 2)(x + 5) (c) (x + 1)(x + 6) (d) (x − 2)(x − 5)
  18. 18.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 904 (b) 784 (c) 1034 (d) 1024
  19. 19.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1764 (b) 1444 (c) 1596 (d) 1454
  20. 20.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 7569 (b) 8109 (c) 8649 (d) 8659
  21. 21.If a + b = 10 and ab = 24, what is a² + b²? (a) 100 (b) 52 (c) 76 (d) 148
  22. 22.What is the product of the roots of x² − 11x + 30 = 0? (a) 30 (b) 31 (c) 29 (d) 11
  23. 23.What are the roots of x² − 9x + 14 = 0? (a) 1 and 14 (b) 2 and 7 (c) 3 and 6 (d) 1 and 8
  24. 24.If a + b = 17 and ab = 70, what is a² + b²? (a) 429 (b) 289 (c) 149 (d) 219
  25. 25.If a + b = 13 and ab = 22, what is a² + b²? (a) 169 (b) 125 (c) 213 (d) 147
  26. 26.If a + b = 17 and ab = 72, what is (a − b)²? (a) 217 (b) 289 (c) 145 (d) 1
  27. 27.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2025 (b) 2475 (c) 2035 (d) 3025
  28. 28.If a + b = 12 and ab = 35, what is a² + b²? (a) 214 (b) 109 (c) 74 (d) 144
  29. 29.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6401 (b) 6571 (c) 6561 (d) 6241
  30. 30.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 56 (c) 16 (d) 15
  31. 31.If a + b = 11 and ab = 10, what is (a − b)²? (a) 121 (b) 101 (c) 81 (d) 9
  32. 32.If a + b = 19 and ab = 88, what is (a − b)²? (a) 3 (b) 185 (c) 361 (d) 9
  33. 33.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x + 6)(x + 8) (c) (x + 1)(x + 13) (d) (x − 6)(x − 8)
  34. 34.If a + b = 7 and ab = 10, what is a² + b²? (a) 49 (b) 29 (c) 39 (d) 69
  35. 35.If a + b = 7 and ab = 6, what is a² + b²? (a) 43 (b) 49 (c) 37 (d) 61
  36. 36.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x − 3)(x − 9) (c) (x + 1)(x + 27) (d) (x + 3)(x + 9)
  37. 37.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8101 (b) 8291 (c) 8281 (d) 7921
  38. 38.If a + b = 8 and ab = 15, what is (a − b)²? (a) 2 (b) 64 (c) 4 (d) 34
  39. 39.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 5476 (b) 4916 (c) 4356 (d) 5486
  40. 40.If a + b = 15 and ab = 54, what is (a − b)²? (a) 117 (b) 3 (c) 225 (d) 9

Answer key

  1. (x + 6)(x + 7)
  2. 3 and 5
  3. 11
  4. (x + 3)(x + 6)
  5. (x + 5)(x + 8)
  6. 4 and 9
  7. 7
  8. 3025
  9. 1849
  10. 68
  11. 12
  12. (x + 4)(x + 7)
  13. 3025
  14. 5
  15. 24
  16. (x + 8)(x + 9)
  17. (x + 2)(x + 5)
  18. 1024
  19. 1444
  20. 8649
  21. 52
  22. 30
  23. 2 and 7
  24. 149
  25. 125
  26. 1
  27. 2025
  28. 74
  29. 6561
  30. 15
  31. 81
  32. 9
  33. (x + 6)(x + 8)
  34. 29
  35. 37
  36. (x + 3)(x + 9)
  37. 8281
  38. 4
  39. 5476
  40. 9

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