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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 250 · 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.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6401 (b) 6241 (c) 6571 (d) 6561
  3. 3.What is the sum of the roots of x² − 9x + 20 = 0? (a) 10 (b) 9 (c) 20 (d) 8
  4. 4.If a + b = 20 and ab = 99, what is (a − b)²? (a) 4 (b) 2 (c) 202 (d) 400
  5. 5.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 63) (c) (x + 1)(x + 15) (d) (x + 7)(x + 9)
  6. 6.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 63 (c) 17 (d) 16
  7. 7.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8116 (b) 8846 (c) 7396 (d) 8836
  8. 8.If a + b = 6 and ab = 5, what is (a − b)²? (a) 36 (b) 26 (c) 4 (d) 16
  9. 9.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 8)(x + 9) (c) (x + 1)(x + 16) (d) (x + 1)(x + 72)
  10. 10.What is the product of the roots of x² − 12x + 27 = 0? (a) 27 (b) 28 (c) 12 (d) 26
  11. 11.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 1)(x + 7) (c) (x + 1)(x + 15) (d) (x + 3)(x + 5)
  12. 12.If a + b = 17 and ab = 70, what is (a − b)²? (a) 289 (b) 3 (c) 9 (d) 149
  13. 13.Which is the correct factorisation of x² + 10x + 24? (a) (x + 4)(x + 6) (b) (x + 1)(x + 9) (c) (x − 4)(x − 6) (d) (x + 1)(x + 24)
  14. 14.Which is the correct factorisation of x² + 13x + 36? (a) (x + 4)(x + 9) (b) (x + 1)(x + 12) (c) (x − 4)(x − 9) (d) (x + 1)(x + 36)
  15. 15.If a + b = 18 and ab = 77, what is a² + b²? (a) 170 (b) 247 (c) 478 (d) 324
  16. 16.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 2 and 36 (c) 1 and 72 (d) 8 and 9
  17. 17.If a + b = 14 and ab = 48, what is (a − b)²? (a) 196 (b) 100 (c) 2 (d) 4
  18. 18.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1089 (b) 909 (c) 1099 (d) 729
  19. 19.What is the product of the roots of x² − 14x + 48 = 0? (a) 14 (b) 49 (c) 47 (d) 48
  20. 20.If a + b = 16 and ab = 63, what is a² + b²? (a) 256 (b) 130 (c) 193 (d) 382
  21. 21.If a + b = 13 and ab = 40, what is (a − b)²? (a) 9 (b) 89 (c) 3 (d) 169
  22. 22.If a + b = 15 and ab = 56, what is (a − b)²? (a) 113 (b) 225 (c) 1 (d) 169
  23. 23.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7235 (b) 5625 (c) 7225 (d) 6425
  24. 24.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x + 1)(x + 28) (c) (x + 4)(x + 7) (d) (x − 4)(x − 7)
  25. 25.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 916 (b) 676 (c) 1156 (d) 1166
  26. 26.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1454 (b) 1596 (c) 1444 (d) 1764
  27. 27.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6734 (b) 6084 (c) 6724 (d) 6404
  28. 28.What is the product of the roots of x² − 10x + 16 = 0? (a) 15 (b) 17 (c) 10 (d) 16
  29. 29.If a + b = 20 and ab = 96, what is (a − b)²? (a) 4 (b) 400 (c) 208 (d) 16
  30. 30.What is the sum of the roots of x² − 8x + 15 = 0? (a) 15 (b) 7 (c) 9 (d) 8
  31. 31.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 11025 (b) 11035 (c) 9025 (d) 10025
  32. 32.If a + b = 13 and ab = 42, what is (a − b)²? (a) 85 (b) 169 (c) 127 (d) 1
  33. 33.If a + b = 17 and ab = 72, what is (a − b)²? (a) 145 (b) 217 (c) 289 (d) 1
  34. 34.What is the sum of the roots of x² − 15x + 54 = 0? (a) 16 (b) 14 (c) 15 (d) 54
  35. 35.What is the sum of the roots of x² − 6x + 8 = 0? (a) 5 (b) 7 (c) 6 (d) 8
  36. 36.If a + b = 10 and ab = 16, what is (a − b)²? (a) 100 (b) 36 (c) 68 (d) 6
  37. 37.If a + b = 11 and ab = 18, what is a² + b²? (a) 157 (b) 121 (c) 85 (d) 103
  38. 38.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 13) (c) (x + 6)(x + 8) (d) (x + 1)(x + 48)
  39. 39.If a + b = 19 and ab = 88, what is a² + b²? (a) 537 (b) 185 (c) 273 (d) 361
  40. 40.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 271 (c) 1 (d) 361

Answer key

  1. (x + 6)(x + 7)
  2. 6561
  3. 9
  4. 4
  5. (x + 7)(x + 9)
  6. 16
  7. 8836
  8. 16
  9. (x + 8)(x + 9)
  10. 27
  11. (x + 3)(x + 5)
  12. 9
  13. (x + 4)(x + 6)
  14. (x + 4)(x + 9)
  15. 170
  16. 8 and 9
  17. 4
  18. 1089
  19. 48
  20. 130
  21. 9
  22. 1
  23. 7225
  24. (x + 4)(x + 7)
  25. 1156
  26. 1444
  27. 6724
  28. 16
  29. 16
  30. 8
  31. 11025
  32. 1
  33. 1
  34. 15
  35. 6
  36. 36
  37. 85
  38. (x + 6)(x + 8)
  39. 185
  40. 1

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