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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 867 · 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 + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x + 1)(x + 16)
  2. 2.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 1)(x + 8) (c) (x + 1)(x + 5) (d) (x + 2)(x + 4)
  3. 3.If a + b = 8 and ab = 15, what is a² + b²? (a) 34 (b) 49 (c) 64 (d) 94
  4. 4.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1225 (b) 2035 (c) 1625 (d) 2025
  5. 5.If a + b = 13 and ab = 12, what is a² + b²? (a) 193 (b) 169 (c) 145 (d) 157
  6. 6.If a + b = 6 and ab = 8, what is a² + b²? (a) 20 (b) 52 (c) 36 (d) 28
  7. 7.What is the product of the roots of x² − 8x + 15 = 0? (a) 14 (b) 8 (c) 16 (d) 15
  8. 8.If a + b = 11 and ab = 24, what is a² + b²? (a) 169 (b) 121 (c) 97 (d) 73
  9. 9.What are the roots of x² − 11x + 24 = 0? (a) 3 and 8 (b) 1 and 10 (c) 2 and 12 (d) 1 and 24
  10. 10.If a + b = 17 and ab = 70, what is (a − b)²? (a) 3 (b) 9 (c) 149 (d) 289
  11. 11.If a + b = 19 and ab = 88, what is (a − b)²? (a) 9 (b) 361 (c) 185 (d) 3
  12. 12.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4356 (b) 4916 (c) 5486 (d) 5476
  13. 13.If a + b = 16 and ab = 63, what is a² + b²? (a) 382 (b) 193 (c) 130 (d) 256
  14. 14.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6391 (b) 5939 (c) 5929 (d) 6889
  15. 15.If a + b = 19 and ab = 88, what is a² + b²? (a) 185 (b) 273 (c) 537 (d) 361
  16. 16.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 225 (c) 441 (d) 657
  17. 17.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 16) (b) (x − 2)(x − 8) (c) (x + 1)(x + 9) (d) (x + 2)(x + 8)
  18. 18.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1601 (b) 1691 (c) 1521 (d) 1681
  19. 19.Which is the correct factorisation of x² + 9x + 14? (a) (x + 2)(x + 7) (b) (x + 1)(x + 8) (c) (x + 1)(x + 14) (d) (x − 2)(x − 7)
  20. 20.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2219 (b) 2209 (c) 2491 (d) 2809
  21. 21.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4366 (b) 4884 (c) 5476 (d) 4356
  22. 22.What is the product of the roots of x² − 11x + 18 = 0? (a) 18 (b) 17 (c) 19 (d) 11
  23. 23.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x + 1)(x + 54) (c) (x + 6)(x + 9) (d) (x − 6)(x − 9)
  24. 24.If a + b = 14 and ab = 33, what is (a − b)²? (a) 8 (b) 196 (c) 130 (d) 64
  25. 25.What is the product of the roots of x² − 6x + 8 = 0? (a) 6 (b) 9 (c) 7 (d) 8
  26. 26.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)
  27. 27.If a + b = 5 and ab = 4, what is (a − b)²? (a) 25 (b) 3 (c) 9 (d) 17
  28. 28.What is the sum of the roots of x² − 5x + 6 = 0? (a) 7 (b) 6 (c) 5 (d) 4
  29. 29.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6241 (b) 6401 (c) 6571 (d) 6561
  30. 30.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x + 4)(x + 5) (c) (x + 1)(x + 8) (d) (x − 4)(x − 5)
  31. 31.If a + b = 17 and ab = 72, what is a² + b²? (a) 289 (b) 145 (c) 217 (d) 433
  32. 32.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 541 (c) 181 (d) 361
  33. 33.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 925 (b) 625 (c) 1235 (d) 1225
  34. 34.If a + b = 15 and ab = 36, what is (a − b)²? (a) 9 (b) 153 (c) 81 (d) 225
  35. 35.What is the product of the roots of x² − 15x + 54 = 0? (a) 55 (b) 53 (c) 15 (d) 54
  36. 36.If a + b = 18 and ab = 77, what is a² + b²? (a) 478 (b) 170 (c) 247 (d) 324
  37. 37.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 7)(x + 8) (c) (x + 1)(x + 56) (d) (x − 7)(x − 8)
  38. 38.If a + b = 14 and ab = 24, what is a² + b²? (a) 244 (b) 196 (c) 172 (d) 148
  39. 39.If a + b = 15 and ab = 36, what is a² + b²? (a) 297 (b) 225 (c) 189 (d) 153
  40. 40.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 484 (b) 324 (c) 396 (d) 334

Answer key

  1. (x + 8)(x + 9)
  2. (x + 2)(x + 4)
  3. 34
  4. 2025
  5. 145
  6. 20
  7. 15
  8. 73
  9. 3 and 8
  10. 9
  11. 9
  12. 5476
  13. 130
  14. 5929
  15. 185
  16. 225
  17. (x + 2)(x + 8)
  18. 1681
  19. (x + 2)(x + 7)
  20. 2209
  21. 4356
  22. 18
  23. (x + 6)(x + 9)
  24. 64
  25. 8
  26. (x + 7)(x + 9)
  27. 9
  28. 5
  29. 6561
  30. (x + 4)(x + 5)
  31. 145
  32. 181
  33. 1225
  34. 81
  35. 54
  36. 170
  37. (x + 7)(x + 8)
  38. 148
  39. 153
  40. 324

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