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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 84 · 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 42²? (a) 1444 (b) 1604 (c) 1774 (d) 1764
  2. 2.What is the product of the roots of x² − 10x + 24 = 0? (a) 10 (b) 24 (c) 25 (d) 23
  3. 3.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4356 (b) 5476 (c) 4884 (d) 4366
  4. 4.If a + b = 20 and ab = 96, what is (a − b)²? (a) 4 (b) 400 (c) 16 (d) 208
  5. 5.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 8) (b) (x − 2)(x − 7) (c) (x + 2)(x + 7) (d) (x + 1)(x + 14)
  6. 6.If a + b = 9 and ab = 18, what is a² + b²? (a) 81 (b) 63 (c) 117 (d) 45
  7. 7.What is the sum of the roots of x² − 11x + 18 = 0? (a) 12 (b) 10 (c) 11 (d) 18
  8. 8.If a + b = 7 and ab = 10, what is a² + b²? (a) 49 (b) 39 (c) 69 (d) 29
  9. 9.If a + b = 12 and ab = 35, what is (a − b)²? (a) 4 (b) 144 (c) 2 (d) 74
  10. 10.If a + b = 14 and ab = 24, what is a² + b²? (a) 244 (b) 148 (c) 172 (d) 196
  11. 11.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x + 1)(x + 16)
  12. 12.What are the roots of x² − 12x + 27 = 0? (a) 2 and 10 (b) 1 and 11 (c) 1 and 27 (d) 3 and 9
  13. 13.If a + b = 15 and ab = 56, what is a² + b²? (a) 337 (b) 113 (c) 225 (d) 169
  14. 14.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6561 (b) 6401 (c) 6571 (d) 6241
  15. 15.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1225 (b) 925 (c) 625 (d) 1235
  16. 16.What is the sum of the roots of x² − 10x + 24 = 0? (a) 10 (b) 11 (c) 9 (d) 24
  17. 17.If a + b = 17 and ab = 72, what is a² + b²? (a) 289 (b) 217 (c) 433 (d) 145
  18. 18.Which is the correct factorisation of x² + 9x + 18? (a) (x + 3)(x + 6) (b) (x − 3)(x − 6) (c) (x + 1)(x + 18) (d) (x + 1)(x + 8)
  19. 19.What is the product of the roots of x² − 12x + 32 = 0? (a) 33 (b) 32 (c) 31 (d) 12
  20. 20.If a + b = 16 and ab = 60, what is a² + b²? (a) 376 (b) 136 (c) 196 (d) 256
  21. 21.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 181 (c) 271 (d) 361
  22. 22.If a + b = 16 and ab = 63, what is (a − b)²? (a) 4 (b) 130 (c) 2 (d) 256
  23. 23.What is the sum of the roots of x² − 11x + 30 = 0? (a) 12 (b) 10 (c) 30 (d) 11
  24. 24.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)
  25. 25.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 48) (c) (x + 6)(x + 8) (d) (x + 1)(x + 13)
  26. 26.If a + b = 13 and ab = 36, what is a² + b²? (a) 97 (b) 169 (c) 241 (d) 133
  27. 27.If a + b = 10 and ab = 21, what is a² + b²? (a) 79 (b) 100 (c) 142 (d) 58
  28. 28.What is the sum of the roots of x² − 13x + 36 = 0? (a) 12 (b) 13 (c) 14 (d) 36
  29. 29.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1859 (b) 1369 (c) 1849 (d) 1609
  30. 30.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 73 (c) 72 (d) 71
  31. 31.Which is the correct factorisation of x² + 11x + 18? (a) (x − 2)(x − 9) (b) (x + 1)(x + 10) (c) (x + 2)(x + 9) (d) (x + 1)(x + 18)
  32. 32.Which is the correct factorisation of x² + 10x + 24? (a) (x + 4)(x + 6) (b) (x + 1)(x + 9) (c) (x + 1)(x + 24) (d) (x − 4)(x − 6)
  33. 33.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4901 (b) 5051 (c) 5041 (d) 4761
  34. 34.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)
  35. 35.If a + b = 14 and ab = 45, what is a² + b²? (a) 196 (b) 286 (c) 151 (d) 106
  36. 36.What is the sum of the roots of x² − 12x + 35 = 0? (a) 13 (b) 35 (c) 11 (d) 12
  37. 37.What are the roots of x² − 11x + 18 = 0? (a) 3 and 6 (b) 2 and 9 (c) 1 and 18 (d) 1 and 10
  38. 38.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3625 (b) 3025 (c) 4235 (d) 4225
  39. 39.What is the product of the roots of x² − 8x + 15 = 0? (a) 14 (b) 16 (c) 15 (d) 8
  40. 40.If a + b = 15 and ab = 56, what is (a − b)²? (a) 169 (b) 1 (c) 113 (d) 225

Answer key

  1. 1764
  2. 24
  3. 4356
  4. 16
  5. (x + 2)(x + 7)
  6. 45
  7. 11
  8. 29
  9. 4
  10. 148
  11. (x + 8)(x + 9)
  12. 3 and 9
  13. 113
  14. 6561
  15. 1225
  16. 10
  17. 145
  18. (x + 3)(x + 6)
  19. 32
  20. 136
  21. 181
  22. 4
  23. 11
  24. (x + 6)(x + 7)
  25. (x + 6)(x + 8)
  26. 97
  27. 58
  28. 13
  29. 1849
  30. 72
  31. (x + 2)(x + 9)
  32. (x + 4)(x + 6)
  33. 5041
  34. (x + 4)(x + 8)
  35. 106
  36. 12
  37. 2 and 9
  38. 4225
  39. 15
  40. 1

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