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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 52 · 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 = 15 and ab = 56, what is (a − b)²? (a) 169 (b) 1 (c) 113 (d) 225
  2. 2.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 676 (b) 884 (c) 686 (d) 1156
  3. 3.If a + b = 9 and ab = 8, what is a² + b²? (a) 73 (b) 65 (c) 97 (d) 81
  4. 4.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1225 (b) 2035 (c) 2025 (d) 1625
  5. 5.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 1)(x + 24) (c) (x + 1)(x + 9) (d) (x + 4)(x + 6)
  6. 6.What is the sum of the roots of x² − 9x + 20 = 0? (a) 9 (b) 20 (c) 8 (d) 10
  7. 7.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x − 7)(x − 9) (c) (x + 7)(x + 9) (d) (x + 1)(x + 15)
  8. 8.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 15 (c) 13 (d) 14
  9. 9.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4225 (b) 4235 (c) 4875 (d) 5625
  10. 10.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 404 (b) 484 (c) 494 (d) 324
  11. 11.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1936 (b) 1306 (c) 1296 (d) 1584
  12. 12.If a + b = 10 and ab = 21, what is (a − b)²? (a) 4 (b) 100 (c) 58 (d) 16
  13. 13.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x + 2)(x + 3) (c) (x − 2)(x − 3) (d) (x + 1)(x + 4)
  14. 14.What is the sum of the roots of x² − 14x + 45 = 0? (a) 14 (b) 45 (c) 15 (d) 13
  15. 15.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 2 and 28 (c) 7 and 8 (d) 1 and 14
  16. 16.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8846 (b) 8836 (c) 7396 (d) 8116
  17. 17.What is the sum of the roots of x² − 8x + 15 = 0? (a) 15 (b) 9 (c) 7 (d) 8
  18. 18.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1691 (b) 1681 (c) 1601 (d) 1521
  19. 19.If a + b = 12 and ab = 32, what is (a − b)²? (a) 16 (b) 4 (c) 144 (d) 80
  20. 20.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)
  21. 21.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x + 1)(x + 8) (c) (x − 4)(x − 5) (d) (x + 1)(x + 20)
  22. 22.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x + 1)(x + 13) (c) (x − 6)(x − 8) (d) (x + 1)(x + 48)
  23. 23.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x − 3)(x − 4) (c) (x + 3)(x + 4) (d) (x + 1)(x + 12)
  24. 24.What is the product of the roots of x² − 12x + 32 = 0? (a) 31 (b) 33 (c) 12 (d) 32
  25. 25.What is the product of the roots of x² − 13x + 36 = 0? (a) 37 (b) 13 (c) 35 (d) 36
  26. 26.What are the roots of x² − 8x + 12 = 0? (a) 1 and 7 (b) 3 and 4 (c) 1 and 12 (d) 2 and 6
  27. 27.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 16 (c) 72 (d) 18
  28. 28.What is the product of the roots of x² − 11x + 18 = 0? (a) 11 (b) 17 (c) 18 (d) 19
  29. 29.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 9025 (b) 9035 (c) 7225 (d) 8125
  30. 30.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6241 (b) 6561 (c) 6571 (d) 6401
  31. 31.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)
  32. 32.Which is the correct factorisation of x² + 7x + 10? (a) (x − 2)(x − 5) (b) (x + 1)(x + 10) (c) (x + 2)(x + 5) (d) (x + 1)(x + 6)
  33. 33.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x − 5)(x − 9) (c) (x + 1)(x + 13) (d) (x + 5)(x + 9)
  34. 34.If a + b = 14 and ab = 48, what is (a − b)²? (a) 2 (b) 100 (c) 4 (d) 196
  35. 35.If a + b = 18 and ab = 72, what is a² + b²? (a) 324 (b) 252 (c) 180 (d) 468
  36. 36.If a + b = 9 and ab = 18, what is a² + b²? (a) 117 (b) 45 (c) 81 (d) 63
  37. 37.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 3)(x + 9) (c) (x + 1)(x + 11) (d) (x + 1)(x + 27)
  38. 38.What are the roots of x² − 11x + 24 = 0? (a) 3 and 8 (b) 1 and 10 (c) 1 and 24 (d) 2 and 12
  39. 39.If a + b = 20 and ab = 96, what is (a − b)²? (a) 400 (b) 208 (c) 4 (d) 16
  40. 40.What is the sum of the roots of x² − 10x + 24 = 0? (a) 24 (b) 9 (c) 10 (d) 11

Answer key

  1. 1
  2. 676
  3. 65
  4. 2025
  5. (x + 4)(x + 6)
  6. 9
  7. (x + 7)(x + 9)
  8. 14
  9. 4225
  10. 484
  11. 1296
  12. 16
  13. (x + 2)(x + 3)
  14. 14
  15. 7 and 8
  16. 8836
  17. 8
  18. 1681
  19. 16
  20. (x + 8)(x + 9)
  21. (x + 4)(x + 5)
  22. (x + 6)(x + 8)
  23. (x + 3)(x + 4)
  24. 32
  25. 36
  26. 2 and 6
  27. 17
  28. 18
  29. 9025
  30. 6561
  31. (x + 4)(x + 8)
  32. (x + 2)(x + 5)
  33. (x + 5)(x + 9)
  34. 4
  35. 180
  36. 45
  37. (x + 3)(x + 9)
  38. 3 and 8
  39. 16
  40. 10

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