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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 999 · 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 − 7x + 12 = 0? (a) 13 (b) 11 (c) 7 (d) 12
  2. 2.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
  3. 3.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x − 4)(x − 5) (c) (x + 1)(x + 20) (d) (x + 4)(x + 5)
  4. 4.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 10025 (b) 9025 (c) 11035 (d) 11025
  5. 5.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 625 (b) 925 (c) 1235 (d) 1225
  6. 6.If a + b = 15 and ab = 36, what is (a − b)²? (a) 153 (b) 225 (c) 9 (d) 81
  7. 7.If a + b = 5 and ab = 4, what is a² + b²? (a) 21 (b) 25 (c) 33 (d) 17
  8. 8.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2916 (b) 2926 (c) 2116 (d) 2516
  9. 9.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x − 5)(x − 8) (c) (x + 5)(x + 8) (d) (x + 1)(x + 12)
  10. 10.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8116 (b) 8846 (c) 8836 (d) 7396
  11. 11.What is the sum of the roots of x² − 6x + 8 = 0? (a) 7 (b) 8 (c) 6 (d) 5
  12. 12.What are the roots of x² − 9x + 18 = 0? (a) 2 and 9 (b) 3 and 6 (c) 1 and 18 (d) 1 and 8
  13. 13.If a + b = 11 and ab = 18, what is a² + b²? (a) 85 (b) 103 (c) 157 (d) 121
  14. 14.If a + b = 10 and ab = 16, what is (a − b)²? (a) 68 (b) 6 (c) 100 (d) 36
  15. 15.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3584 (b) 4096 (c) 3146 (d) 3136
  16. 16.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7235 (b) 5625 (c) 7225 (d) 6425
  17. 17.If a + b = 6 and ab = 8, what is (a − b)²? (a) 36 (b) 2 (c) 4 (d) 20
  18. 18.If a + b = 17 and ab = 70, what is a² + b²? (a) 149 (b) 429 (c) 289 (d) 219
  19. 19.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 8)(x + 9) (c) (x + 1)(x + 16) (d) (x − 8)(x − 9)
  20. 20.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 7)(x + 9) (c) (x + 1)(x + 63) (d) (x + 1)(x + 15)
  21. 21.If a + b = 9 and ab = 20, what is a² + b²? (a) 121 (b) 61 (c) 41 (d) 81
  22. 22.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6889 (b) 5929 (c) 5939 (d) 6391
  23. 23.If a + b = 7 and ab = 12, what is (a − b)²? (a) 49 (b) 25 (c) 1 (d) 37
  24. 24.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x + 2)(x + 4) (c) (x − 2)(x − 4) (d) (x + 1)(x + 8)
  25. 25.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 13) (c) (x + 1)(x + 48) (d) (x + 6)(x + 8)
  26. 26.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 8) (b) (x + 2)(x + 7) (c) (x − 2)(x − 7) (d) (x + 1)(x + 14)
  27. 27.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2401 (b) 2501 (c) 2611 (d) 2601
  28. 28.What are the roots of x² − 6x + 8 = 0? (a) 1 and 5 (b) 1 and 8 (c) 2 and 5 (d) 2 and 4
  29. 29.What is the product of the roots of x² − 11x + 28 = 0? (a) 27 (b) 29 (c) 28 (d) 11
  30. 30.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 841 (b) 961 (c) 901 (d) 971
  31. 31.What is the sum of the roots of x² − 11x + 18 = 0? (a) 11 (b) 12 (c) 18 (d) 10
  32. 32.If a + b = 18 and ab = 80, what is a² + b²? (a) 484 (b) 244 (c) 324 (d) 164
  33. 33.If a + b = 14 and ab = 48, what is a² + b²? (a) 196 (b) 100 (c) 292 (d) 148
  34. 34.If a + b = 17 and ab = 72, what is a² + b²? (a) 217 (b) 433 (c) 145 (d) 289
  35. 35.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1379 (b) 1849 (c) 1591 (d) 1369
  36. 36.What is the sum of the roots of x² − 8x + 15 = 0? (a) 8 (b) 9 (c) 7 (d) 15
  37. 37.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7066 (b) 7056 (c) 5776 (d) 6416
  38. 38.If a + b = 16 and ab = 55, what is (a − b)²? (a) 6 (b) 256 (c) 36 (d) 146
  39. 39.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 16 (c) 17 (d) 72
  40. 40.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1609 (b) 1859 (c) 1849 (d) 1369

Answer key

  1. 12
  2. 6 and 7
  3. (x + 4)(x + 5)
  4. 11025
  5. 1225
  6. 81
  7. 17
  8. 2916
  9. (x + 5)(x + 8)
  10. 8836
  11. 6
  12. 3 and 6
  13. 85
  14. 36
  15. 3136
  16. 7225
  17. 4
  18. 149
  19. (x + 8)(x + 9)
  20. (x + 7)(x + 9)
  21. 41
  22. 5929
  23. 1
  24. (x + 2)(x + 4)
  25. (x + 6)(x + 8)
  26. (x + 2)(x + 7)
  27. 2601
  28. 2 and 4
  29. 28
  30. 961
  31. 11
  32. 164
  33. 100
  34. 145
  35. 1369
  36. 8
  37. 7056
  38. 36
  39. 17
  40. 1849

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