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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 124 · 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 97²? (a) 10609 (b) 9991 (c) 9419 (d) 9409
  2. 2.If a + b = 12 and ab = 27, what is a² + b²? (a) 117 (b) 198 (c) 144 (d) 90
  3. 3.What are the roots of x² − 8x + 15 = 0? (a) 1 and 15 (b) 1 and 7 (c) 3 and 5 (d) 2 and 6
  4. 4.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 16 (d) 18
  5. 5.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 901 (b) 841 (c) 971 (d) 961
  6. 6.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1625 (b) 2035 (c) 2025 (d) 1225
  7. 7.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)
  8. 8.If a + b = 14 and ab = 45, what is (a − b)²? (a) 196 (b) 16 (c) 4 (d) 106
  9. 9.If a + b = 5 and ab = 6, what is a² + b²? (a) 37 (b) 13 (c) 25 (d) 19
  10. 10.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 9409 (b) 10609 (c) 10619 (d) 10009
  11. 11.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x + 1)(x + 11) (c) (x + 4)(x + 8) (d) (x − 4)(x − 8)
  12. 12.If a + b = 16 and ab = 55, what is a² + b²? (a) 256 (b) 146 (c) 201 (d) 366
  13. 13.If a + b = 13 and ab = 40, what is a² + b²? (a) 249 (b) 169 (c) 129 (d) 89
  14. 14.If a + b = 14 and ab = 45, what is a² + b²? (a) 151 (b) 196 (c) 286 (d) 106
  15. 15.If a + b = 9 and ab = 20, what is a² + b²? (a) 61 (b) 121 (c) 81 (d) 41
  16. 16.If a + b = 19 and ab = 90, what is a² + b²? (a) 181 (b) 361 (c) 541 (d) 271
  17. 17.What is the sum of the roots of x² − 8x + 12 = 0? (a) 9 (b) 7 (c) 8 (d) 12
  18. 18.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x − 5)(x − 6) (c) (x + 5)(x + 6) (d) (x + 1)(x + 10)
  19. 19.What are the roots of x² − 11x + 30 = 0? (a) 2 and 15 (b) 1 and 30 (c) 5 and 6 (d) 1 and 10
  20. 20.What are the roots of x² − 12x + 32 = 0? (a) 4 and 8 (b) 1 and 11 (c) 1 and 32 (d) 2 and 16
  21. 21.If a + b = 16 and ab = 60, what is a² + b²? (a) 256 (b) 136 (c) 196 (d) 376
  22. 22.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 72) (c) (x − 8)(x − 9) (d) (x + 1)(x + 16)
  23. 23.If a + b = 10 and ab = 16, what is (a − b)²? (a) 36 (b) 68 (c) 100 (d) 6
  24. 24.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 361 (b) 441 (c) 371 (d) 399
  25. 25.If a + b = 7 and ab = 10, what is a² + b²? (a) 69 (b) 49 (c) 29 (d) 39
  26. 26.If a + b = 19 and ab = 84, what is a² + b²? (a) 529 (b) 361 (c) 193 (d) 277
  27. 27.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x − 4)(x − 5) (c) (x + 4)(x + 5) (d) (x + 1)(x + 8)
  28. 28.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x + 1)(x + 15) (c) (x + 3)(x + 5) (d) (x − 3)(x − 5)
  29. 29.If a + b = 4 and ab = 3, what is a² + b²? (a) 16 (b) 10 (c) 22 (d) 13
  30. 30.If a + b = 14 and ab = 48, what is a² + b²? (a) 148 (b) 292 (c) 100 (d) 196
  31. 31.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 1 and 16 (c) 2 and 36 (d) 8 and 9
  32. 32.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1034 (b) 904 (c) 784 (d) 1024
  33. 33.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 324 (b) 396 (c) 334 (d) 484
  34. 34.If a + b = 20 and ab = 99, what is (a − b)²? (a) 202 (b) 4 (c) 2 (d) 400
  35. 35.If a + b = 15 and ab = 56, what is a² + b²? (a) 225 (b) 169 (c) 337 (d) 113
  36. 36.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1936 (b) 1306 (c) 1584 (d) 1296
  37. 37.If a + b = 17 and ab = 60, what is (a − b)²? (a) 7 (b) 49 (c) 169 (d) 289
  38. 38.What is the product of the roots of x² − 7x + 12 = 0? (a) 7 (b) 13 (c) 11 (d) 12
  39. 39.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 441 (b) 401 (c) 361 (d) 451
  40. 40.If a + b = 11 and ab = 28, what is (a − b)²? (a) 65 (b) 3 (c) 121 (d) 9

Answer key

  1. 9409
  2. 90
  3. 3 and 5
  4. 17
  5. 961
  6. 2025
  7. (x + 5)(x + 8)
  8. 16
  9. 13
  10. 10609
  11. (x + 4)(x + 8)
  12. 146
  13. 89
  14. 106
  15. 41
  16. 181
  17. 8
  18. (x + 5)(x + 6)
  19. 5 and 6
  20. 4 and 8
  21. 136
  22. (x + 8)(x + 9)
  23. 36
  24. 361
  25. 29
  26. 193
  27. (x + 4)(x + 5)
  28. (x + 3)(x + 5)
  29. 10
  30. 100
  31. 8 and 9
  32. 1024
  33. 324
  34. 4
  35. 113
  36. 1296
  37. 49
  38. 12
  39. 441
  40. 9

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