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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 166 · 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.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 17 (c) 16 (d) 72
  2. 2.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 225 (b) 425 (c) 635 (d) 625
  3. 3.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 7579 (b) 7569 (c) 8091 (d) 8649
  4. 4.What is the product of the roots of x² − 10x + 21 = 0? (a) 22 (b) 10 (c) 20 (d) 21
  5. 5.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2496 (b) 2704 (c) 2314 (d) 2304
  6. 6.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)
  7. 7.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 1225 (b) 635 (c) 625 (d) 875
  8. 8.If a + b = 15 and ab = 36, what is (a − b)²? (a) 81 (b) 9 (c) 153 (d) 225
  9. 9.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x − 6)(x − 8) (c) (x + 6)(x + 8) (d) (x + 1)(x + 13)
  10. 10.What is the sum of the roots of x² − 11x + 18 = 0? (a) 18 (b) 10 (c) 12 (d) 11
  11. 11.If a + b = 11 and ab = 18, what is (a − b)²? (a) 85 (b) 7 (c) 121 (d) 49
  12. 12.What is the sum of the roots of x² − 10x + 24 = 0? (a) 9 (b) 11 (c) 24 (d) 10
  13. 13.What is the sum of the roots of x² − 7x + 10 = 0? (a) 8 (b) 6 (c) 7 (d) 10
  14. 14.If a + b = 18 and ab = 80, what is a² + b²? (a) 244 (b) 324 (c) 484 (d) 164
  15. 15.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 409 (b) 539 (c) 289 (d) 529
  16. 16.What are the roots of x² − 9x + 20 = 0? (a) 1 and 8 (b) 4 and 5 (c) 1 and 20 (d) 2 and 10
  17. 17.If a + b = 10 and ab = 21, what is (a − b)²? (a) 58 (b) 16 (c) 100 (d) 4
  18. 18.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2714 (b) 2704 (c) 2304 (d) 2504
  19. 19.If a + b = 15 and ab = 56, what is a² + b²? (a) 113 (b) 337 (c) 225 (d) 169
  20. 20.What is the sum of the roots of x² − 13x + 40 = 0? (a) 13 (b) 12 (c) 40 (d) 14
  21. 21.Which is the correct factorisation of x² + 8x + 12? (a) (x − 2)(x − 6) (b) (x + 2)(x + 6) (c) (x + 1)(x + 7) (d) (x + 1)(x + 12)
  22. 22.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 13 (c) 15 (d) 14
  23. 23.If a + b = 19 and ab = 84, what is (a − b)²? (a) 5 (b) 361 (c) 193 (d) 25
  24. 24.If a + b = 17 and ab = 60, what is (a − b)²? (a) 289 (b) 7 (c) 49 (d) 169
  25. 25.If a + b = 14 and ab = 45, what is a² + b²? (a) 286 (b) 151 (c) 196 (d) 106
  26. 26.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1225 (b) 1235 (c) 625 (d) 925
  27. 27.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 401 (b) 451 (c) 441 (d) 361
  28. 28.What are the roots of x² − 10x + 16 = 0? (a) 3 and 7 (b) 1 and 9 (c) 1 and 16 (d) 2 and 8
  29. 29.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)
  30. 30.What are the roots of x² − 11x + 28 = 0? (a) 1 and 28 (b) 1 and 10 (c) 4 and 7 (d) 2 and 14
  31. 31.If a + b = 3 and ab = 2, what is a² + b²? (a) 5 (b) 9 (c) 7 (d) 13
  32. 32.Which is the correct factorisation of x² + 10x + 16? (a) (x + 2)(x + 8) (b) (x + 1)(x + 16) (c) (x − 2)(x − 8) (d) (x + 1)(x + 9)
  33. 33.If a + b = 12 and ab = 35, what is a² + b²? (a) 214 (b) 109 (c) 144 (d) 74
  34. 34.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6889 (b) 5929 (c) 6899 (d) 6409
  35. 35.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 5939 (b) 6391 (c) 6889 (d) 5929
  36. 36.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1584 (b) 1306 (c) 1936 (d) 1296
  37. 37.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x − 2)(x − 4) (c) (x + 1)(x + 5) (d) (x + 2)(x + 4)
  38. 38.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3721 (b) 3481 (c) 3601 (d) 3731
  39. 39.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x + 1)(x + 45) (c) (x − 5)(x − 9) (d) (x + 5)(x + 9)
  40. 40.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8281 (b) 7921 (c) 8101 (d) 8291

Answer key

  1. 17
  2. 625
  3. 7569
  4. 21
  5. 2304
  6. (x + 8)(x + 9)
  7. 625
  8. 81
  9. (x + 6)(x + 8)
  10. 11
  11. 49
  12. 10
  13. 7
  14. 164
  15. 529
  16. 4 and 5
  17. 16
  18. 2704
  19. 113
  20. 13
  21. (x + 2)(x + 6)
  22. 14
  23. 25
  24. 49
  25. 106
  26. 1225
  27. 441
  28. 2 and 8
  29. (x + 5)(x + 6)
  30. 4 and 7
  31. 5
  32. (x + 2)(x + 8)
  33. 74
  34. 6889
  35. 5929
  36. 1296
  37. (x + 2)(x + 4)
  38. 3721
  39. (x + 5)(x + 9)
  40. 8281

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