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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 811 · 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.What is the sum of the roots of x² − 9x + 18 = 0? (a) 9 (b) 8 (c) 18 (d) 10
  2. 2.If a + b = 16 and ab = 48, what is (a − b)²? (a) 256 (b) 8 (c) 64 (d) 160
  3. 3.What is the sum of the roots of x² − 8x + 12 = 0? (a) 9 (b) 8 (c) 7 (d) 12
  4. 4.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 5)(x + 6) (c) (x − 5)(x − 6) (d) (x + 1)(x + 30)
  5. 5.Which is the correct factorisation of x² + 11x + 24? (a) (x + 3)(x + 8) (b) (x + 1)(x + 24) (c) (x − 3)(x − 8) (d) (x + 1)(x + 10)
  6. 6.Which is the correct factorisation of x² + 13x + 40? (a) (x + 5)(x + 8) (b) (x − 5)(x − 8) (c) (x + 1)(x + 12) (d) (x + 1)(x + 40)
  7. 7.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 71 (c) 17 (d) 72
  8. 8.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x + 3)(x + 7) (c) (x + 1)(x + 21) (d) (x − 3)(x − 7)
  9. 9.If a + b = 18 and ab = 77, what is (a − b)²? (a) 170 (b) 324 (c) 16 (d) 4
  10. 10.If a + b = 11 and ab = 18, what is (a − b)²? (a) 85 (b) 7 (c) 121 (d) 49
  11. 11.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3584 (b) 4096 (c) 3146 (d) 3136
  12. 12.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4634 (b) 4896 (c) 4624 (d) 5184
  13. 13.Which is the correct factorisation of x² + 11x + 18? (a) (x + 2)(x + 9) (b) (x − 2)(x − 9) (c) (x + 1)(x + 10) (d) (x + 1)(x + 18)
  14. 14.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x + 1)(x + 45) (c) (x + 5)(x + 9) (d) (x − 5)(x − 9)
  15. 15.What are the roots of x² − 15x + 54 = 0? (a) 2 and 27 (b) 6 and 9 (c) 1 and 14 (d) 1 and 54
  16. 16.What is the sum of the roots of x² − 13x + 40 = 0? (a) 14 (b) 40 (c) 12 (d) 13
  17. 17.If a + b = 9 and ab = 14, what is (a − b)²? (a) 5 (b) 25 (c) 81 (d) 53
  18. 18.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2491 (b) 2209 (c) 2809 (d) 2219
  19. 19.If a + b = 16 and ab = 63, what is a² + b²? (a) 382 (b) 193 (c) 256 (d) 130
  20. 20.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 409 (b) 289 (c) 529 (d) 539
  21. 21.If a + b = 9 and ab = 18, what is (a − b)²? (a) 9 (b) 3 (c) 81 (d) 45
  22. 22.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1089 (b) 1099 (c) 909 (d) 729
  23. 23.If a + b = 8 and ab = 15, what is a² + b²? (a) 49 (b) 94 (c) 34 (d) 64
  24. 24.What is the sum of the roots of x² − 8x + 15 = 0? (a) 9 (b) 7 (c) 15 (d) 8
  25. 25.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 5776 (b) 6416 (c) 7056 (d) 7066
  26. 26.What is the sum of the roots of x² − 15x + 54 = 0? (a) 14 (b) 54 (c) 16 (d) 15
  27. 27.What is the sum of the roots of x² − 6x + 8 = 0? (a) 7 (b) 5 (c) 8 (d) 6
  28. 28.What are the roots of x² − 12x + 32 = 0? (a) 2 and 16 (b) 1 and 32 (c) 4 and 8 (d) 1 and 11
  29. 29.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x + 4)(x + 5) (c) (x − 4)(x − 5) (d) (x + 1)(x + 20)
  30. 30.If a + b = 15 and ab = 50, what is a² + b²? (a) 175 (b) 325 (c) 125 (d) 225
  31. 31.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)
  32. 32.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 16) (b) (x − 2)(x − 8) (c) (x + 1)(x + 9) (d) (x + 2)(x + 8)
  33. 33.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 657 (c) 441 (d) 225
  34. 34.What is the product of the roots of x² − 9x + 14 = 0? (a) 13 (b) 9 (c) 14 (d) 15
  35. 35.If a + b = 15 and ab = 36, what is (a − b)²? (a) 9 (b) 153 (c) 225 (d) 81
  36. 36.If a + b = 4 and ab = 3, what is (a − b)²? (a) 10 (b) 2 (c) 16 (d) 4
  37. 37.If a + b = 6 and ab = 8, what is a² + b²? (a) 52 (b) 28 (c) 36 (d) 20
  38. 38.What is the product of the roots of x² − 8x + 15 = 0? (a) 16 (b) 15 (c) 14 (d) 8
  39. 39.If a + b = 3 and ab = 2, what is (a − b)²? (a) 5 (b) 7 (c) 9 (d) 1
  40. 40.If a + b = 12 and ab = 27, what is (a − b)²? (a) 144 (b) 6 (c) 36 (d) 90

Answer key

  1. 9
  2. 64
  3. 8
  4. (x + 5)(x + 6)
  5. (x + 3)(x + 8)
  6. (x + 5)(x + 8)
  7. 72
  8. (x + 3)(x + 7)
  9. 16
  10. 49
  11. 3136
  12. 4624
  13. (x + 2)(x + 9)
  14. (x + 5)(x + 9)
  15. 6 and 9
  16. 13
  17. 25
  18. 2209
  19. 130
  20. 529
  21. 9
  22. 1089
  23. 34
  24. 8
  25. 7056
  26. 15
  27. 6
  28. 4 and 8
  29. (x + 4)(x + 5)
  30. 125
  31. (x + 8)(x + 9)
  32. (x + 2)(x + 8)
  33. 225
  34. 14
  35. 81
  36. 4
  37. 20
  38. 15
  39. 1
  40. 36

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