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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 174 · 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 + 14x + 45? (a) (x + 1)(x + 13) (b) (x + 5)(x + 9) (c) (x + 1)(x + 45) (d) (x − 5)(x − 9)
  2. 2.What are the roots of x² − 6x + 8 = 0? (a) 1 and 8 (b) 1 and 5 (c) 2 and 5 (d) 2 and 4
  3. 3.If a + b = 11 and ab = 10, what is a² + b²? (a) 111 (b) 141 (c) 101 (d) 121
  4. 4.If a + b = 17 and ab = 70, what is (a − b)²? (a) 3 (b) 289 (c) 9 (d) 149
  5. 5.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 15) (c) (x + 1)(x + 63) (d) (x + 7)(x + 9)
  6. 6.If a + b = 14 and ab = 45, what is a² + b²? (a) 151 (b) 286 (c) 106 (d) 196
  7. 7.If a + b = 9 and ab = 8, what is a² + b²? (a) 73 (b) 65 (c) 81 (d) 97
  8. 8.If a + b = 16 and ab = 63, what is (a − b)²? (a) 130 (b) 2 (c) 4 (d) 256
  9. 9.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x − 6)(x − 9) (c) (x + 6)(x + 9) (d) (x + 1)(x + 14)
  10. 10.If a + b = 15 and ab = 36, what is (a − b)²? (a) 81 (b) 153 (c) 9 (d) 225
  11. 11.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 73 (c) 71 (d) 17
  12. 12.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 14) (c) (x + 7)(x + 8) (d) (x + 1)(x + 56)
  13. 13.What is the product of the roots of x² − 7x + 10 = 0? (a) 11 (b) 9 (c) 10 (d) 7
  14. 14.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 441 (b) 371 (c) 399 (d) 361
  15. 15.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1849 (b) 1591 (c) 1369 (d) 1379
  16. 16.What are the roots of x² − 12x + 35 = 0? (a) 2 and 10 (b) 1 and 35 (c) 1 and 11 (d) 5 and 7
  17. 17.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1681 (b) 1531 (c) 1599 (d) 1521
  18. 18.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x + 1)(x + 40) (c) (x + 5)(x + 8) (d) (x − 5)(x − 8)
  19. 19.What is the product of the roots of x² − 7x + 12 = 0? (a) 7 (b) 12 (c) 13 (d) 11
  20. 20.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3035 (b) 4225 (c) 3575 (d) 3025
  21. 21.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9975 (b) 9025 (c) 11025 (d) 9035
  22. 22.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x − 2)(x − 6) (c) (x + 1)(x + 12) (d) (x + 2)(x + 6)
  23. 23.If a + b = 6 and ab = 5, what is (a − b)²? (a) 4 (b) 16 (c) 36 (d) 26
  24. 24.If a + b = 6 and ab = 8, what is a² + b²? (a) 36 (b) 28 (c) 52 (d) 20
  25. 25.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 5184 (b) 4634 (c) 4896 (d) 4624
  26. 26.If a + b = 17 and ab = 72, what is (a − b)²? (a) 217 (b) 145 (c) 1 (d) 289
  27. 27.What is the sum of the roots of x² − 12x + 32 = 0? (a) 32 (b) 13 (c) 12 (d) 11
  28. 28.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 1)(x + 10) (c) (x − 5)(x − 6) (d) (x + 5)(x + 6)
  29. 29.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1609 (b) 1859 (c) 1369 (d) 1849
  30. 30.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x − 2)(x − 4) (c) (x + 1)(x + 8) (d) (x + 2)(x + 4)
  31. 31.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)
  32. 32.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 16 (c) 18 (d) 72
  33. 33.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)
  34. 34.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 657 (c) 225 (d) 333
  35. 35.If a + b = 11 and ab = 18, what is (a − b)²? (a) 85 (b) 121 (c) 49 (d) 7
  36. 36.What is the sum of the roots of x² − 5x + 6 = 0? (a) 4 (b) 5 (c) 6 (d) 7
  37. 37.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 7225 (b) 9035 (c) 8125 (d) 9025
  38. 38.What is the product of the roots of x² − 11x + 30 = 0? (a) 30 (b) 29 (c) 31 (d) 11
  39. 39.What is the product of the roots of x² − 13x + 42 = 0? (a) 42 (b) 43 (c) 13 (d) 41
  40. 40.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 181 (c) 541 (d) 361

Answer key

  1. (x + 5)(x + 9)
  2. 2 and 4
  3. 101
  4. 9
  5. (x + 7)(x + 9)
  6. 106
  7. 65
  8. 4
  9. (x + 6)(x + 9)
  10. 81
  11. 72
  12. (x + 7)(x + 8)
  13. 10
  14. 361
  15. 1369
  16. 5 and 7
  17. 1521
  18. (x + 5)(x + 8)
  19. 12
  20. 3025
  21. 9025
  22. (x + 2)(x + 6)
  23. 16
  24. 20
  25. 4624
  26. 1
  27. 12
  28. (x + 5)(x + 6)
  29. 1849
  30. (x + 2)(x + 4)
  31. (x + 3)(x + 6)
  32. 17
  33. (x + 2)(x + 7)
  34. 225
  35. 49
  36. 5
  37. 9025
  38. 30
  39. 42
  40. 181

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