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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 542 · 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.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x − 4)(x − 7) (c) (x + 1)(x + 10) (d) (x + 4)(x + 7)
  2. 2.Which is the correct factorisation of x² + 13x + 36? (a) (x + 4)(x + 9) (b) (x + 1)(x + 36) (c) (x + 1)(x + 12) (d) (x − 4)(x − 9)
  3. 3.If a + b = 8 and ab = 12, what is a² + b²? (a) 64 (b) 88 (c) 52 (d) 40
  4. 4.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 6384 (b) 5776 (c) 5786 (d) 7056
  5. 5.What are the roots of x² − 13x + 42 = 0? (a) 1 and 42 (b) 1 and 12 (c) 2 and 21 (d) 6 and 7
  6. 6.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8091 (b) 7579 (c) 8649 (d) 7569
  7. 7.What is the sum of the roots of x² − 11x + 30 = 0? (a) 11 (b) 30 (c) 10 (d) 12
  8. 8.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 896 (b) 1024 (c) 794 (d) 784
  9. 9.What are the roots of x² − 8x + 15 = 0? (a) 3 and 5 (b) 2 and 6 (c) 1 and 7 (d) 1 and 15
  10. 10.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x − 2)(x − 5) (c) (x + 1)(x + 10) (d) (x + 2)(x + 5)
  11. 11.If a + b = 16 and ab = 63, what is (a − b)²? (a) 4 (b) 256 (c) 2 (d) 130
  12. 12.If a + b = 12 and ab = 35, what is a² + b²? (a) 109 (b) 74 (c) 214 (d) 144
  13. 13.What is the sum of the roots of x² − 13x + 42 = 0? (a) 13 (b) 42 (c) 12 (d) 14
  14. 14.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 1)(x + 12) (c) (x − 6)(x − 7) (d) (x + 6)(x + 7)
  15. 15.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 7)(x + 8) (c) (x + 1)(x + 56) (d) (x + 1)(x + 14)
  16. 16.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6899 (b) 6409 (c) 6889 (d) 5929
  17. 17.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 361 (b) 441 (c) 399 (d) 371
  18. 18.If a + b = 6 and ab = 5, what is (a − b)²? (a) 36 (b) 4 (c) 26 (d) 16
  19. 19.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 7225 (b) 8125 (c) 9025 (d) 9035
  20. 20.If a + b = 19 and ab = 88, what is a² + b²? (a) 273 (b) 537 (c) 361 (d) 185
  21. 21.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1034 (b) 784 (c) 1024 (d) 904
  22. 22.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)
  23. 23.What are the roots of x² − 9x + 14 = 0? (a) 3 and 6 (b) 1 and 8 (c) 2 and 7 (d) 1 and 14
  24. 24.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x + 4)(x + 8) (c) (x + 1)(x + 11) (d) (x − 4)(x − 8)
  25. 25.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 899 (b) 851 (c) 961 (d) 841
  26. 26.If a + b = 8 and ab = 7, what is a² + b²? (a) 57 (b) 50 (c) 78 (d) 64
  27. 27.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 1 and 72 (c) 1 and 16 (d) 2 and 36
  28. 28.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x + 1)(x + 20) (c) (x + 1)(x + 8) (d) (x − 4)(x − 5)
  29. 29.If a + b = 15 and ab = 50, what is a² + b²? (a) 175 (b) 325 (c) 225 (d) 125
  30. 30.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 225 (b) 375 (c) 625 (d) 235
  31. 31.If a + b = 13 and ab = 42, what is a² + b²? (a) 253 (b) 127 (c) 169 (d) 85
  32. 32.If a + b = 21 and ab = 108, what is a² + b²? (a) 225 (b) 657 (c) 441 (d) 333
  33. 33.If a + b = 20 and ab = 96, what is (a − b)²? (a) 208 (b) 16 (c) 400 (d) 4
  34. 34.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)
  35. 35.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 1)(x + 72) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)
  36. 36.What is the sum of the roots of x² − 9x + 18 = 0? (a) 9 (b) 18 (c) 8 (d) 10
  37. 37.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 15 (c) 56 (d) 16
  38. 38.If a + b = 19 and ab = 84, what is a² + b²? (a) 361 (b) 277 (c) 193 (d) 529
  39. 39.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x + 1)(x + 7) (c) (x − 3)(x − 5) (d) (x + 3)(x + 5)
  40. 40.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 16 (d) 18

Answer key

  1. (x + 4)(x + 7)
  2. (x + 4)(x + 9)
  3. 40
  4. 5776
  5. 6 and 7
  6. 7569
  7. 11
  8. 784
  9. 3 and 5
  10. (x + 2)(x + 5)
  11. 4
  12. 74
  13. 13
  14. (x + 6)(x + 7)
  15. (x + 7)(x + 8)
  16. 6889
  17. 361
  18. 16
  19. 9025
  20. 185
  21. 1024
  22. (x + 7)(x + 9)
  23. 2 and 7
  24. (x + 4)(x + 8)
  25. 841
  26. 50
  27. 8 and 9
  28. (x + 4)(x + 5)
  29. 125
  30. 225
  31. 85
  32. 225
  33. 16
  34. (x + 3)(x + 8)
  35. (x + 8)(x + 9)
  36. 9
  37. 15
  38. 193
  39. (x + 3)(x + 5)
  40. 17

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