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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 217 · 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 = 16 and ab = 63, what is a² + b²? (a) 193 (b) 256 (c) 382 (d) 130
  2. 2.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 5)(x + 9) (c) (x + 1)(x + 45) (d) (x + 1)(x + 13)
  3. 3.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)
  4. 4.If a + b = 13 and ab = 36, what is (a − b)²? (a) 97 (b) 5 (c) 169 (d) 25
  5. 5.If a + b = 12 and ab = 35, what is (a − b)²? (a) 74 (b) 2 (c) 4 (d) 144
  6. 6.What are the roots of x² − 14x + 48 = 0? (a) 6 and 8 (b) 2 and 24 (c) 1 and 13 (d) 1 and 48
  7. 7.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 657 (c) 225 (d) 441
  8. 8.If a + b = 14 and ab = 45, what is (a − b)²? (a) 106 (b) 196 (c) 4 (d) 16
  9. 9.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 9 (c) 3 (d) 441
  10. 10.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x − 6)(x − 7) (c) (x + 1)(x + 12) (d) (x + 1)(x + 42)
  11. 11.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1596 (b) 1764 (c) 1454 (d) 1444
  12. 12.If a + b = 15 and ab = 50, what is (a − b)²? (a) 125 (b) 5 (c) 25 (d) 225
  13. 13.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)
  14. 14.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x + 1)(x + 6) (c) (x − 2)(x − 5) (d) (x + 2)(x + 5)
  15. 15.If a + b = 16 and ab = 48, what is (a − b)²? (a) 64 (b) 160 (c) 8 (d) 256
  16. 16.What is the product of the roots of x² − 15x + 54 = 0? (a) 55 (b) 54 (c) 15 (d) 53
  17. 17.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9025 (b) 9975 (c) 9035 (d) 11025
  18. 18.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8836 (b) 8116 (c) 7396 (d) 8846
  19. 19.If a + b = 18 and ab = 72, what is (a − b)²? (a) 324 (b) 6 (c) 36 (d) 180
  20. 20.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x + 1)(x + 12) (c) (x − 5)(x − 8) (d) (x + 5)(x + 8)
  21. 21.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 271 (c) 181 (d) 541
  22. 22.If a + b = 3 and ab = 2, what is a² + b²? (a) 7 (b) 9 (c) 5 (d) 13
  23. 23.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4875 (b) 5625 (c) 4225 (d) 4235
  24. 24.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)
  25. 25.What is the sum of the roots of x² − 10x + 24 = 0? (a) 9 (b) 11 (c) 24 (d) 10
  26. 26.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 676 (b) 686 (c) 1156 (d) 884
  27. 27.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 361 (b) 451 (c) 401 (d) 441
  28. 28.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7744 (b) 8096 (c) 8464 (d) 7754
  29. 29.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3575 (b) 3025 (c) 4225 (d) 3035
  30. 30.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6561 (b) 6401 (c) 6571 (d) 6241
  31. 31.What is the sum of the roots of x² − 12x + 32 = 0? (a) 32 (b) 12 (c) 13 (d) 11
  32. 32.What are the roots of x² − 13x + 42 = 0? (a) 6 and 7 (b) 1 and 42 (c) 1 and 12 (d) 2 and 21
  33. 33.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)
  34. 34.If a + b = 10 and ab = 16, what is a² + b²? (a) 84 (b) 68 (c) 100 (d) 132
  35. 35.What is the sum of the roots of x² − 8x + 15 = 0? (a) 15 (b) 7 (c) 8 (d) 9
  36. 36.What is the sum of the roots of x² − 13x + 36 = 0? (a) 14 (b) 13 (c) 12 (d) 36
  37. 37.If a + b = 16 and ab = 48, what is a² + b²? (a) 160 (b) 256 (c) 208 (d) 352
  38. 38.What is the product of the roots of x² − 11x + 24 = 0? (a) 24 (b) 23 (c) 25 (d) 11
  39. 39.If a + b = 8 and ab = 15, what is a² + b²? (a) 64 (b) 49 (c) 94 (d) 34
  40. 40.What is the sum of the roots of x² − 8x + 12 = 0? (a) 7 (b) 8 (c) 12 (d) 9

Answer key

  1. 130
  2. (x + 5)(x + 9)
  3. (x + 7)(x + 8)
  4. 25
  5. 4
  6. 6 and 8
  7. 225
  8. 16
  9. 9
  10. (x + 6)(x + 7)
  11. 1444
  12. 25
  13. (x + 2)(x + 4)
  14. (x + 2)(x + 5)
  15. 64
  16. 54
  17. 9025
  18. 8836
  19. 36
  20. (x + 5)(x + 8)
  21. 181
  22. 5
  23. 4225
  24. (x + 5)(x + 6)
  25. 10
  26. 676
  27. 441
  28. 7744
  29. 3025
  30. 6561
  31. 12
  32. 6 and 7
  33. (x + 8)(x + 9)
  34. 68
  35. 8
  36. 13
  37. 160
  38. 24
  39. 34
  40. 8

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