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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 559 · 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.What is the product of the roots of x² − 12x + 27 = 0? (a) 27 (b) 28 (c) 12 (d) 26
  2. 2.What is the product of the roots of x² − 13x + 42 = 0? (a) 43 (b) 13 (c) 42 (d) 41
  3. 3.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8474 (b) 8104 (c) 8464 (d) 7744
  4. 4.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)
  5. 5.If a + b = 18 and ab = 72, what is a² + b²? (a) 324 (b) 468 (c) 252 (d) 180
  6. 6.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1774 (b) 1444 (c) 1764 (d) 1604
  7. 7.What is the sum of the roots of x² − 10x + 24 = 0? (a) 9 (b) 11 (c) 10 (d) 24
  8. 8.What is the product of the roots of x² − 11x + 28 = 0? (a) 28 (b) 27 (c) 29 (d) 11
  9. 9.What are the roots of x² − 15x + 54 = 0? (a) 1 and 54 (b) 1 and 14 (c) 6 and 9 (d) 2 and 27
  10. 10.What are the roots of x² − 10x + 24 = 0? (a) 4 and 6 (b) 1 and 24 (c) 1 and 9 (d) 2 and 12
  11. 11.What is the product of the roots of x² − 10x + 16 = 0? (a) 15 (b) 16 (c) 17 (d) 10
  12. 12.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 271 (c) 1 (d) 361
  13. 13.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x + 3)(x + 5) (c) (x − 3)(x − 5) (d) (x + 1)(x + 15)
  14. 14.If a + b = 9 and ab = 20, what is a² + b²? (a) 81 (b) 41 (c) 121 (d) 61
  15. 15.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)
  16. 16.What is the sum of the roots of x² − 14x + 45 = 0? (a) 14 (b) 13 (c) 15 (d) 45
  17. 17.If a + b = 15 and ab = 54, what is (a − b)²? (a) 117 (b) 3 (c) 9 (d) 225
  18. 18.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 7921 (b) 8291 (c) 8101 (d) 8281
  19. 19.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6084 (b) 6734 (c) 6404 (d) 6724
  20. 20.If a + b = 12 and ab = 27, what is (a − b)²? (a) 36 (b) 144 (c) 6 (d) 90
  21. 21.Which is the correct factorisation of x² + 11x + 28? (a) (x + 4)(x + 7) (b) (x − 4)(x − 7) (c) (x + 1)(x + 28) (d) (x + 1)(x + 10)
  22. 22.What is the sum of the roots of x² − 8x + 15 = 0? (a) 9 (b) 8 (c) 7 (d) 15
  23. 23.If a + b = 17 and ab = 60, what is a² + b²? (a) 289 (b) 409 (c) 169 (d) 229
  24. 24.If a + b = 14 and ab = 45, what is (a − b)²? (a) 16 (b) 196 (c) 106 (d) 4
  25. 25.If a + b = 10 and ab = 24, what is a² + b²? (a) 76 (b) 100 (c) 52 (d) 148
  26. 26.What is the sum of the roots of x² − 11x + 28 = 0? (a) 10 (b) 12 (c) 28 (d) 11
  27. 27.What are the roots of x² − 13x + 42 = 0? (a) 1 and 42 (b) 6 and 7 (c) 2 and 21 (d) 1 and 12
  28. 28.What are the roots of x² − 11x + 30 = 0? (a) 2 and 15 (b) 1 and 10 (c) 5 and 6 (d) 1 and 30
  29. 29.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x − 5)(x − 6) (c) (x + 1)(x + 30) (d) (x + 1)(x + 10)
  30. 30.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x − 5)(x − 8) (c) (x + 5)(x + 8) (d) (x + 1)(x + 40)
  31. 31.If a + b = 17 and ab = 72, what is (a − b)²? (a) 289 (b) 217 (c) 1 (d) 145
  32. 32.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 72 (c) 17 (d) 73
  33. 33.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)
  34. 34.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2525 (b) 3025 (c) 2025 (d) 3035
  35. 35.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 6)(x + 7) (c) (x + 1)(x + 42) (d) (x + 1)(x + 12)
  36. 36.If a + b = 18 and ab = 77, what is a² + b²? (a) 170 (b) 478 (c) 247 (d) 324
  37. 37.If a + b = 11 and ab = 28, what is (a − b)²? (a) 9 (b) 65 (c) 121 (d) 3
  38. 38.What are the roots of x² − 14x + 48 = 0? (a) 1 and 13 (b) 2 and 24 (c) 6 and 8 (d) 1 and 48
  39. 39.Which is the correct factorisation of x² + 6x + 8? (a) (x + 2)(x + 4) (b) (x − 2)(x − 4) (c) (x + 1)(x + 5) (d) (x + 1)(x + 8)
  40. 40.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x + 1)(x + 45) (c) (x − 5)(x − 9) (d) (x + 1)(x + 13)

Answer key

  1. 27
  2. 42
  3. 8464
  4. (x + 4)(x + 8)
  5. 180
  6. 1764
  7. 10
  8. 28
  9. 6 and 9
  10. 4 and 6
  11. 16
  12. 1
  13. (x + 3)(x + 5)
  14. 41
  15. (x + 8)(x + 9)
  16. 14
  17. 9
  18. 8281
  19. 6724
  20. 36
  21. (x + 4)(x + 7)
  22. 8
  23. 169
  24. 16
  25. 52
  26. 11
  27. 6 and 7
  28. 5 and 6
  29. (x + 5)(x + 6)
  30. (x + 5)(x + 8)
  31. 1
  32. 72
  33. (x + 3)(x + 8)
  34. 3025
  35. (x + 6)(x + 7)
  36. 170
  37. 9
  38. 6 and 8
  39. (x + 2)(x + 4)
  40. (x + 5)(x + 9)

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