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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 760 · 40 questions
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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.If a + b = 17 and ab = 66, what is a² + b²? (a) 223 (b) 157 (c) 289 (d) 421
  2. 2.If a + b = 9 and ab = 18, what is a² + b²? (a) 45 (b) 81 (c) 63 (d) 117
  3. 3.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x − 6)(x − 8) (c) (x + 6)(x + 8) (d) (x + 1)(x + 48)
  4. 4.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 9604 (b) 10414 (c) 10004 (d) 10404
  5. 5.Which is the correct factorisation of x² + 9x + 14? (a) (x + 2)(x + 7) (b) (x + 1)(x + 14) (c) (x + 1)(x + 8) (d) (x − 2)(x − 7)
  6. 6.If a + b = 14 and ab = 48, what is (a − b)²? (a) 196 (b) 2 (c) 100 (d) 4
  7. 7.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 391 (b) 289 (c) 299 (d) 529
  8. 8.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 17 (d) 16
  9. 9.If a + b = 5 and ab = 6, what is a² + b²? (a) 37 (b) 13 (c) 19 (d) 25
  10. 10.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1575 (b) 1225 (c) 2025 (d) 1235
  11. 11.If a + b = 11 and ab = 18, what is a² + b²? (a) 121 (b) 157 (c) 85 (d) 103
  12. 12.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x − 3)(x − 7) (c) (x + 3)(x + 7) (d) (x + 1)(x + 21)
  13. 13.If a + b = 14 and ab = 48, what is a² + b²? (a) 100 (b) 148 (c) 292 (d) 196
  14. 14.What is the product of the roots of x² − 13x + 40 = 0? (a) 13 (b) 39 (c) 41 (d) 40
  15. 15.What is the product of the roots of x² − 16x + 63 = 0? (a) 16 (b) 64 (c) 63 (d) 62
  16. 16.If a + b = 15 and ab = 54, what is a² + b²? (a) 333 (b) 117 (c) 225 (d) 171
  17. 17.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1849 (b) 1379 (c) 1369 (d) 1591
  18. 18.What is the product of the roots of x² − 12x + 32 = 0? (a) 31 (b) 33 (c) 32 (d) 12
  19. 19.What is the sum of the roots of x² − 14x + 48 = 0? (a) 15 (b) 48 (c) 14 (d) 13
  20. 20.What is the sum of the roots of x² − 13x + 42 = 0? (a) 14 (b) 42 (c) 13 (d) 12
  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.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 181 (c) 361 (d) 541
  23. 23.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)
  24. 24.If a + b = 14 and ab = 45, what is (a − b)²? (a) 196 (b) 16 (c) 106 (d) 4
  25. 25.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x + 1)(x + 20) (c) (x + 4)(x + 5) (d) (x − 4)(x − 5)
  26. 26.What are the roots of x² − 13x + 42 = 0? (a) 1 and 12 (b) 2 and 21 (c) 1 and 42 (d) 6 and 7
  27. 27.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 57 (c) 55 (d) 56
  28. 28.If a + b = 15 and ab = 44, what is a² + b²? (a) 225 (b) 313 (c) 137 (d) 181
  29. 29.If a + b = 12 and ab = 27, what is a² + b²? (a) 117 (b) 144 (c) 90 (d) 198
  30. 30.What is the sum of the roots of x² − 10x + 21 = 0? (a) 21 (b) 10 (c) 9 (d) 11
  31. 31.If a + b = 7 and ab = 10, what is (a − b)²? (a) 29 (b) 49 (c) 9 (d) 3
  32. 32.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4499 (b) 4489 (c) 5329 (d) 4891
  33. 33.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 441 (b) 399 (c) 371 (d) 361
  34. 34.If a + b = 16 and ab = 60, what is a² + b²? (a) 376 (b) 256 (c) 196 (d) 136
  35. 35.What is the sum of the roots of x² − 10x + 16 = 0? (a) 11 (b) 10 (c) 16 (d) 9
  36. 36.What is the product of the roots of x² − 9x + 18 = 0? (a) 19 (b) 17 (c) 18 (d) 9
  37. 37.What is the product of the roots of x² − 13x + 36 = 0? (a) 13 (b) 37 (c) 35 (d) 36
  38. 38.What is the sum of the roots of x² − 11x + 30 = 0? (a) 30 (b) 11 (c) 10 (d) 12
  39. 39.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1306 (b) 1936 (c) 1584 (d) 1296
  40. 40.If a + b = 14 and ab = 33, what is a² + b²? (a) 163 (b) 196 (c) 262 (d) 130

Answer key

  1. 157
  2. 45
  3. (x + 6)(x + 8)
  4. 10404
  5. (x + 2)(x + 7)
  6. 4
  7. 289
  8. 17
  9. 13
  10. 1225
  11. 85
  12. (x + 3)(x + 7)
  13. 100
  14. 40
  15. 63
  16. 117
  17. 1369
  18. 32
  19. 14
  20. 13
  21. (x + 4)(x + 7)
  22. 181
  23. (x + 8)(x + 9)
  24. 16
  25. (x + 4)(x + 5)
  26. 6 and 7
  27. 56
  28. 137
  29. 90
  30. 10
  31. 9
  32. 4489
  33. 361
  34. 136
  35. 10
  36. 18
  37. 36
  38. 11
  39. 1296
  40. 130

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