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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 486 · 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 − 7x + 12 = 0? (a) 12 (b) 8 (c) 6 (d) 7
  2. 2.If a + b = 11 and ab = 10, what is (a − b)²? (a) 101 (b) 121 (c) 81 (d) 9
  3. 3.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 1)(x + 36) (c) (x + 4)(x + 9) (d) (x + 1)(x + 12)
  4. 4.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x − 5)(x − 7) (c) (x + 1)(x + 11) (d) (x + 5)(x + 7)
  5. 5.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x + 1)(x + 8) (c) (x − 4)(x − 5) (d) (x + 4)(x + 5)
  6. 6.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 901 (b) 961 (c) 971 (d) 841
  7. 7.What is the sum of the roots of x² − 11x + 24 = 0? (a) 10 (b) 12 (c) 11 (d) 24
  8. 8.If a + b = 16 and ab = 63, what is (a − b)²? (a) 4 (b) 2 (c) 256 (d) 130
  9. 9.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1369 (b) 1379 (c) 1591 (d) 1849
  10. 10.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 5486 (b) 4356 (c) 4916 (d) 5476
  11. 11.What is the sum of the roots of x² − 8x + 15 = 0? (a) 8 (b) 9 (c) 15 (d) 7
  12. 12.What is the product of the roots of x² − 11x + 18 = 0? (a) 19 (b) 11 (c) 17 (d) 18
  13. 13.What are the roots of x² − 7x + 12 = 0? (a) 1 and 6 (b) 2 and 6 (c) 1 and 12 (d) 3 and 4
  14. 14.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3146 (b) 3584 (c) 4096 (d) 3136
  15. 15.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 6)(x + 8) (c) (x + 1)(x + 48) (d) (x − 6)(x − 8)
  16. 16.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x + 3)(x + 8) (c) (x − 3)(x − 8) (d) (x + 1)(x + 10)
  17. 17.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x + 1)(x + 18) (c) (x − 3)(x − 6) (d) (x + 3)(x + 6)
  18. 18.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 11025 (b) 9975 (c) 9025 (d) 9035
  19. 19.If a + b = 15 and ab = 50, what is a² + b²? (a) 325 (b) 225 (c) 175 (d) 125
  20. 20.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 11) (b) (x + 4)(x + 8) (c) (x − 4)(x − 8) (d) (x + 1)(x + 32)
  21. 21.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
  22. 22.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 404 (b) 324 (c) 484 (d) 494
  23. 23.What is the sum of the roots of x² − 9x + 14 = 0? (a) 8 (b) 9 (c) 10 (d) 14
  24. 24.If a + b = 18 and ab = 80, what is a² + b²? (a) 324 (b) 484 (c) 164 (d) 244
  25. 25.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)
  26. 26.Which is the correct factorisation of x² + 9x + 14? (a) (x + 2)(x + 7) (b) (x + 1)(x + 8) (c) (x − 2)(x − 7) (d) (x + 1)(x + 14)
  27. 27.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 1)(x + 15) (c) (x + 3)(x + 5) (d) (x + 1)(x + 7)
  28. 28.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 5929 (b) 6899 (c) 6409 (d) 6889
  29. 29.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 5184 (b) 4624 (c) 4904 (d) 5194
  30. 30.If a + b = 10 and ab = 21, what is (a − b)²? (a) 100 (b) 16 (c) 58 (d) 4
  31. 31.If a + b = 5 and ab = 6, what is a² + b²? (a) 19 (b) 37 (c) 13 (d) 25
  32. 32.If a + b = 13 and ab = 42, what is (a − b)²? (a) 1 (b) 169 (c) 127 (d) 85
  33. 33.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 55 (c) 57 (d) 56
  34. 34.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7744 (b) 8096 (c) 7754 (d) 8464
  35. 35.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 3035 (b) 2025 (c) 2525 (d) 3025
  36. 36.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 401 (b) 361 (c) 441 (d) 451
  37. 37.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1946 (b) 1616 (c) 1936 (d) 1296
  38. 38.If a + b = 13 and ab = 40, what is a² + b²? (a) 89 (b) 129 (c) 169 (d) 249
  39. 39.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 425 (b) 635 (c) 625 (d) 225
  40. 40.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 71 (c) 72 (d) 73

Answer key

  1. 7
  2. 81
  3. (x + 4)(x + 9)
  4. (x + 5)(x + 7)
  5. (x + 4)(x + 5)
  6. 961
  7. 11
  8. 4
  9. 1369
  10. 5476
  11. 8
  12. 18
  13. 3 and 4
  14. 3136
  15. (x + 6)(x + 8)
  16. (x + 3)(x + 8)
  17. (x + 3)(x + 6)
  18. 9025
  19. 125
  20. (x + 4)(x + 8)
  21. 8 and 9
  22. 484
  23. 9
  24. 164
  25. (x + 8)(x + 9)
  26. (x + 2)(x + 7)
  27. (x + 3)(x + 5)
  28. 6889
  29. 5184
  30. 16
  31. 13
  32. 1
  33. 56
  34. 7744
  35. 3025
  36. 441
  37. 1936
  38. 89
  39. 625
  40. 72

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