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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 352 · 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.If a + b = 7 and ab = 10, what is (a − b)²? (a) 3 (b) 29 (c) 49 (d) 9
  2. 2.If a + b = 19 and ab = 84, what is (a − b)²? (a) 5 (b) 25 (c) 193 (d) 361
  3. 3.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 57 (c) 15 (d) 56
  4. 4.What is the sum of the roots of x² − 5x + 6 = 0? (a) 4 (b) 7 (c) 6 (d) 5
  5. 5.Which is the correct factorisation of x² + 10x + 21? (a) (x − 3)(x − 7) (b) (x + 1)(x + 9) (c) (x + 3)(x + 7) (d) (x + 1)(x + 21)
  6. 6.If a + b = 12 and ab = 32, what is a² + b²? (a) 80 (b) 112 (c) 144 (d) 208
  7. 7.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)
  8. 8.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 1)(x + 7) (c) (x + 1)(x + 15) (d) (x + 3)(x + 5)
  9. 9.What are the roots of x² − 13x + 42 = 0? (a) 2 and 21 (b) 6 and 7 (c) 1 and 42 (d) 1 and 12
  10. 10.What are the roots of x² − 11x + 30 = 0? (a) 2 and 15 (b) 1 and 10 (c) 5 and 6 (d) 1 and 30
  11. 11.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 5184 (b) 4624 (c) 4904 (d) 5194
  12. 12.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 539 (b) 289 (c) 529 (d) 409
  13. 13.Which is the correct factorisation of x² + 7x + 12? (a) (x + 3)(x + 4) (b) (x + 1)(x + 12) (c) (x + 1)(x + 6) (d) (x − 3)(x − 4)
  14. 14.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 925 (b) 625 (c) 1235 (d) 1225
  15. 15.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1444 (b) 1764 (c) 1454 (d) 1596
  16. 16.What is the sum of the roots of x² − 10x + 21 = 0? (a) 11 (b) 10 (c) 9 (d) 21
  17. 17.If a + b = 7 and ab = 6, what is (a − b)²? (a) 49 (b) 5 (c) 25 (d) 37
  18. 18.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2401 (b) 2499 (c) 2601 (d) 2411
  19. 19.What is the sum of the roots of x² − 11x + 30 = 0? (a) 30 (b) 11 (c) 12 (d) 10
  20. 20.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 891 (b) 1089 (c) 729 (d) 739
  21. 21.What is the product of the roots of x² − 14x + 48 = 0? (a) 49 (b) 47 (c) 14 (d) 48
  22. 22.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 1 (c) 361 (d) 271
  23. 23.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2601 (b) 2401 (c) 2501 (d) 2611
  24. 24.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 1)(x + 56) (c) (x + 7)(x + 8) (d) (x − 7)(x − 8)
  25. 25.If a + b = 12 and ab = 11, what is (a − b)²? (a) 100 (b) 122 (c) 10 (d) 144
  26. 26.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10211 (b) 10201 (c) 10001 (d) 9801
  27. 27.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4901 (b) 4761 (c) 5051 (d) 5041
  28. 28.If a + b = 6 and ab = 8, what is a² + b²? (a) 36 (b) 28 (c) 52 (d) 20
  29. 29.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 494 (b) 484 (c) 324 (d) 404
  30. 30.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1369 (b) 1859 (c) 1849 (d) 1609
  31. 31.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)
  32. 32.If a + b = 11 and ab = 24, what is (a − b)²? (a) 121 (b) 5 (c) 73 (d) 25
  33. 33.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 15) (c) (x − 7)(x − 9) (d) (x + 1)(x + 63)
  34. 34.What are the roots of x² − 12x + 27 = 0? (a) 1 and 11 (b) 1 and 27 (c) 3 and 9 (d) 2 and 10
  35. 35.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 625 (b) 235 (c) 375 (d) 225
  36. 36.What is the product of the roots of x² − 13x + 40 = 0? (a) 40 (b) 41 (c) 39 (d) 13
  37. 37.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 5)(x + 6) (c) (x + 1)(x + 30) (d) (x − 5)(x − 6)
  38. 38.If a + b = 21 and ab = 108, what is (a − b)²? (a) 9 (b) 3 (c) 441 (d) 225
  39. 39.What is the sum of the roots of x² − 9x + 20 = 0? (a) 9 (b) 20 (c) 8 (d) 10
  40. 40.If a + b = 15 and ab = 50, what is (a − b)²? (a) 5 (b) 25 (c) 125 (d) 225

Answer key

  1. 9
  2. 25
  3. 56
  4. 5
  5. (x + 3)(x + 7)
  6. 80
  7. (x + 8)(x + 9)
  8. (x + 3)(x + 5)
  9. 6 and 7
  10. 5 and 6
  11. 5184
  12. 529
  13. (x + 3)(x + 4)
  14. 1225
  15. 1444
  16. 10
  17. 25
  18. 2401
  19. 11
  20. 729
  21. 48
  22. 1
  23. 2601
  24. (x + 7)(x + 8)
  25. 100
  26. 10201
  27. 5041
  28. 20
  29. 484
  30. 1849
  31. (x + 4)(x + 5)
  32. 25
  33. (x + 7)(x + 9)
  34. 3 and 9
  35. 225
  36. 40
  37. (x + 5)(x + 6)
  38. 9
  39. 9
  40. 25

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