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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 991 · 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 product of the roots of x² − 14x + 45 = 0? (a) 46 (b) 14 (c) 44 (d) 45
  2. 2.What are the roots of x² − 10x + 21 = 0? (a) 3 and 7 (b) 2 and 8 (c) 1 and 21 (d) 1 and 9
  3. 3.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 1)(x + 48) (c) (x − 6)(x − 8) (d) (x + 6)(x + 8)
  4. 4.If a + b = 16 and ab = 55, what is a² + b²? (a) 201 (b) 366 (c) 256 (d) 146
  5. 5.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 5184 (b) 4624 (c) 4634 (d) 4896
  6. 6.If a + b = 7 and ab = 12, what is a² + b²? (a) 37 (b) 49 (c) 25 (d) 73
  7. 7.If a + b = 14 and ab = 40, what is a² + b²? (a) 196 (b) 276 (c) 156 (d) 116
  8. 8.If a + b = 17 and ab = 60, what is (a − b)²? (a) 49 (b) 7 (c) 169 (d) 289
  9. 9.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7066 (b) 5776 (c) 6416 (d) 7056
  10. 10.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 73 (c) 71 (d) 72
  11. 11.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6889 (b) 5929 (c) 5939 (d) 6391
  12. 12.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x + 5)(x + 7) (c) (x − 5)(x − 7) (d) (x + 1)(x + 11)
  13. 13.Which is the correct factorisation of x² + 9x + 18? (a) (x + 3)(x + 6) (b) (x + 1)(x + 8) (c) (x − 3)(x − 6) (d) (x + 1)(x + 18)
  14. 14.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)
  15. 15.What is the sum of the roots of x² − 13x + 42 = 0? (a) 13 (b) 14 (c) 42 (d) 12
  16. 16.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 416 (c) 586 (d) 576
  17. 17.Which is the correct factorisation of x² + 7x + 10? (a) (x − 2)(x − 5) (b) (x + 1)(x + 10) (c) (x + 2)(x + 5) (d) (x + 1)(x + 6)
  18. 18.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3854 (b) 3364 (c) 3604 (d) 3844
  19. 19.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 56) (c) (x + 7)(x + 8) (d) (x + 1)(x + 14)
  20. 20.What are the roots of x² − 9x + 18 = 0? (a) 1 and 8 (b) 3 and 6 (c) 1 and 18 (d) 2 and 9
  21. 21.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2401 (b) 2501 (c) 2601 (d) 2611
  22. 22.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x − 3)(x − 7) (c) (x + 1)(x + 9) (d) (x + 3)(x + 7)
  23. 23.If a + b = 8 and ab = 12, what is (a − b)²? (a) 64 (b) 4 (c) 40 (d) 16
  24. 24.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2601 (b) 2411 (c) 2401 (d) 2499
  25. 25.What are the roots of x² − 16x + 63 = 0? (a) 1 and 63 (b) 1 and 15 (c) 3 and 21 (d) 7 and 9
  26. 26.What is the product of the roots of x² − 13x + 36 = 0? (a) 13 (b) 37 (c) 36 (d) 35
  27. 27.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 384 (b) 256 (c) 576 (d) 266
  28. 28.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1601 (b) 1681 (c) 1521 (d) 1691
  29. 29.If a + b = 17 and ab = 72, what is a² + b²? (a) 289 (b) 145 (c) 217 (d) 433
  30. 30.Which is the correct factorisation of x² + 11x + 18? (a) (x + 2)(x + 9) (b) (x − 2)(x − 9) (c) (x + 1)(x + 18) (d) (x + 1)(x + 10)
  31. 31.If a + b = 17 and ab = 70, what is (a − b)²? (a) 289 (b) 149 (c) 3 (d) 9
  32. 32.If a + b = 13 and ab = 40, what is a² + b²? (a) 129 (b) 249 (c) 89 (d) 169
  33. 33.What is the sum of the roots of x² − 7x + 12 = 0? (a) 8 (b) 7 (c) 6 (d) 12
  34. 34.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6094 (b) 6084 (c) 6396 (d) 6724
  35. 35.If a + b = 18 and ab = 80, what is a² + b²? (a) 164 (b) 324 (c) 244 (d) 484
  36. 36.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 401 (b) 451 (c) 441 (d) 361
  37. 37.If a + b = 11 and ab = 30, what is (a − b)²? (a) 121 (b) 1 (c) 91 (d) 61
  38. 38.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 10025 (b) 9025 (c) 11035 (d) 11025
  39. 39.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6561 (b) 6241 (c) 6571 (d) 6401
  40. 40.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 1)(x + 42) (c) (x − 6)(x − 7) (d) (x + 6)(x + 7)

Answer key

  1. 45
  2. 3 and 7
  3. (x + 6)(x + 8)
  4. 146
  5. 4624
  6. 25
  7. 116
  8. 49
  9. 7056
  10. 72
  11. 5929
  12. (x + 5)(x + 7)
  13. (x + 3)(x + 6)
  14. (x + 8)(x + 9)
  15. 13
  16. 576
  17. (x + 2)(x + 5)
  18. 3844
  19. (x + 7)(x + 8)
  20. 3 and 6
  21. 2601
  22. (x + 3)(x + 7)
  23. 16
  24. 2401
  25. 7 and 9
  26. 36
  27. 256
  28. 1681
  29. 145
  30. (x + 2)(x + 9)
  31. 9
  32. 89
  33. 7
  34. 6084
  35. 164
  36. 441
  37. 1
  38. 11025
  39. 6561
  40. (x + 6)(x + 7)

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