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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 129 · 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 are the roots of x² − 9x + 14 = 0? (a) 3 and 6 (b) 1 and 14 (c) 1 and 8 (d) 2 and 7
  2. 2.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 5329 (b) 4891 (c) 4499 (d) 4489
  3. 3.What is the sum of the roots of x² − 8x + 15 = 0? (a) 15 (b) 7 (c) 9 (d) 8
  4. 4.What is the product of the roots of x² − 10x + 21 = 0? (a) 22 (b) 21 (c) 20 (d) 10
  5. 5.If a + b = 10 and ab = 9, what is (a − b)²? (a) 82 (b) 100 (c) 64 (d) 8
  6. 6.Which is the correct factorisation of x² + 6x + 8? (a) (x + 2)(x + 4) (b) (x − 2)(x − 4) (c) (x + 1)(x + 8) (d) (x + 1)(x + 5)
  7. 7.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x − 4)(x − 6) (c) (x + 1)(x + 9) (d) (x + 4)(x + 6)
  8. 8.What is the sum of the roots of x² − 15x + 54 = 0? (a) 15 (b) 54 (c) 14 (d) 16
  9. 9.If a + b = 8 and ab = 15, what is (a − b)²? (a) 64 (b) 34 (c) 2 (d) 4
  10. 10.If a + b = 8 and ab = 7, what is a² + b²? (a) 78 (b) 57 (c) 64 (d) 50
  11. 11.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1774 (b) 1444 (c) 1764 (d) 1604
  12. 12.What are the roots of x² − 8x + 12 = 0? (a) 1 and 7 (b) 2 and 6 (c) 3 and 4 (d) 1 and 12
  13. 13.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)
  14. 14.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 6)(x + 8) (c) (x − 6)(x − 8) (d) (x + 1)(x + 48)
  15. 15.If a + b = 9 and ab = 18, what is a² + b²? (a) 63 (b) 117 (c) 81 (d) 45
  16. 16.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2491 (b) 2219 (c) 2809 (d) 2209
  17. 17.What are the roots of x² − 13x + 42 = 0? (a) 6 and 7 (b) 1 and 12 (c) 2 and 21 (d) 1 and 42
  18. 18.What is the sum of the roots of x² − 8x + 12 = 0? (a) 8 (b) 12 (c) 7 (d) 9
  19. 19.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x − 4)(x − 5) (c) (x + 4)(x + 5) (d) (x + 1)(x + 8)
  20. 20.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)
  21. 21.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4235 (b) 4875 (c) 4225 (d) 5625
  22. 22.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x + 1)(x + 28) (c) (x + 4)(x + 7) (d) (x − 4)(x − 7)
  23. 23.If a + b = 3 and ab = 2, what is (a − b)²? (a) 5 (b) 7 (c) 1 (d) 9
  24. 24.What are the roots of x² − 15x + 54 = 0? (a) 1 and 14 (b) 6 and 9 (c) 2 and 27 (d) 1 and 54
  25. 25.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x + 1)(x + 16) (c) (x − 2)(x − 8) (d) (x + 2)(x + 8)
  26. 26.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 7)(x + 8) (c) (x + 1)(x + 56) (d) (x − 7)(x − 8)
  27. 27.If a + b = 12 and ab = 11, what is a² + b²? (a) 122 (b) 133 (c) 166 (d) 144
  28. 28.If a + b = 8 and ab = 15, what is a² + b²? (a) 34 (b) 49 (c) 94 (d) 64
  29. 29.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 657 (c) 333 (d) 225
  30. 30.What is the sum of the roots of x² − 13x + 42 = 0? (a) 14 (b) 13 (c) 12 (d) 42
  31. 31.What is the product of the roots of x² − 11x + 30 = 0? (a) 30 (b) 11 (c) 29 (d) 31
  32. 32.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 1)(x + 30) (c) (x + 5)(x + 6) (d) (x − 5)(x − 6)
  33. 33.Which is the correct factorisation of x² + 5x + 6? (a) (x + 2)(x + 3) (b) (x + 1)(x + 6) (c) (x + 1)(x + 4) (d) (x − 2)(x − 3)
  34. 34.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 7744 (b) 8474 (c) 8464 (d) 8104
  35. 35.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 16 (c) 17 (d) 18
  36. 36.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 7)(x + 9) (c) (x + 1)(x + 63) (d) (x − 7)(x − 9)
  37. 37.What is the product of the roots of x² − 7x + 12 = 0? (a) 7 (b) 13 (c) 12 (d) 11
  38. 38.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2116 (b) 2916 (c) 2126 (d) 2484
  39. 39.Which is the correct factorisation of x² + 12x + 27? (a) (x + 3)(x + 9) (b) (x + 1)(x + 11) (c) (x − 3)(x − 9) (d) (x + 1)(x + 27)
  40. 40.If a + b = 14 and ab = 40, what is (a − b)²? (a) 6 (b) 116 (c) 36 (d) 196

Answer key

  1. 2 and 7
  2. 4489
  3. 8
  4. 21
  5. 64
  6. (x + 2)(x + 4)
  7. (x + 4)(x + 6)
  8. 15
  9. 4
  10. 50
  11. 1764
  12. 2 and 6
  13. (x + 5)(x + 8)
  14. (x + 6)(x + 8)
  15. 45
  16. 2209
  17. 6 and 7
  18. 8
  19. (x + 4)(x + 5)
  20. (x + 8)(x + 9)
  21. 4225
  22. (x + 4)(x + 7)
  23. 1
  24. 6 and 9
  25. (x + 2)(x + 8)
  26. (x + 7)(x + 8)
  27. 122
  28. 34
  29. 225
  30. 13
  31. 30
  32. (x + 5)(x + 6)
  33. (x + 2)(x + 3)
  34. 8464
  35. 17
  36. (x + 7)(x + 9)
  37. 12
  38. 2116
  39. (x + 3)(x + 9)
  40. 36

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