← Identities, polynomials and quadratic roots lesson New set →

Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 331 · 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 − 7x + 10 = 0? (a) 8 (b) 6 (c) 10 (d) 7
  2. 2.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 1)(x + 15) (c) (x + 1)(x + 7) (d) (x + 3)(x + 5)
  3. 3.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 5041 (b) 4761 (c) 4771 (d) 4899
  4. 4.If a + b = 16 and ab = 63, what is a² + b²? (a) 256 (b) 130 (c) 193 (d) 382
  5. 5.What are the roots of x² − 14x + 45 = 0? (a) 1 and 45 (b) 5 and 9 (c) 3 and 15 (d) 1 and 13
  6. 6.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 10025 (b) 11025 (c) 9025 (d) 11035
  7. 7.If a + b = 9 and ab = 14, what is (a − b)²? (a) 53 (b) 81 (c) 25 (d) 5
  8. 8.What is the product of the roots of x² − 14x + 48 = 0? (a) 48 (b) 14 (c) 49 (d) 47
  9. 9.What is the product of the roots of x² − 7x + 12 = 0? (a) 11 (b) 13 (c) 7 (d) 12
  10. 10.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10009 (b) 9409 (c) 10619 (d) 10609
  11. 11.If a + b = 16 and ab = 55, what is a² + b²? (a) 256 (b) 366 (c) 146 (d) 201
  12. 12.What is the sum of the roots of x² − 5x + 6 = 0? (a) 7 (b) 6 (c) 5 (d) 4
  13. 13.If a + b = 11 and ab = 28, what is a² + b²? (a) 121 (b) 65 (c) 93 (d) 177
  14. 14.If a + b = 14 and ab = 40, what is (a − b)²? (a) 6 (b) 36 (c) 116 (d) 196
  15. 15.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
  16. 16.If a + b = 6 and ab = 5, what is (a − b)²? (a) 36 (b) 16 (c) 4 (d) 26
  17. 17.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 73 (c) 17 (d) 71
  18. 18.If a + b = 11 and ab = 24, what is a² + b²? (a) 169 (b) 121 (c) 97 (d) 73
  19. 19.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 289 (c) 145 (d) 217
  20. 20.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)
  21. 21.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1946 (b) 1616 (c) 1936 (d) 1296
  22. 22.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 16) (b) (x − 2)(x − 8) (c) (x + 2)(x + 8) (d) (x + 1)(x + 9)
  23. 23.What are the roots of x² − 10x + 21 = 0? (a) 1 and 9 (b) 2 and 8 (c) 3 and 7 (d) 1 and 21
  24. 24.If a + b = 20 and ab = 96, what is (a − b)²? (a) 4 (b) 400 (c) 16 (d) 208
  25. 25.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 11) (c) (x + 4)(x + 8) (d) (x + 1)(x + 32)
  26. 26.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 7225 (b) 5625 (c) 5635 (d) 6375
  27. 27.What are the roots of x² − 15x + 56 = 0? (a) 7 and 8 (b) 1 and 56 (c) 1 and 14 (d) 2 and 28
  28. 28.If a + b = 7 and ab = 10, what is (a − b)²? (a) 29 (b) 9 (c) 3 (d) 49
  29. 29.What is the product of the roots of x² − 12x + 35 = 0? (a) 36 (b) 12 (c) 34 (d) 35
  30. 30.What are the roots of x² − 8x + 15 = 0? (a) 3 and 5 (b) 1 and 7 (c) 1 and 15 (d) 2 and 6
  31. 31.What is the product of the roots of x² − 13x + 40 = 0? (a) 13 (b) 39 (c) 40 (d) 41
  32. 32.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x + 2)(x + 7) (c) (x + 1)(x + 8) (d) (x − 2)(x − 7)
  33. 33.What is the product of the roots of x² − 9x + 20 = 0? (a) 20 (b) 9 (c) 19 (d) 21
  34. 34.What is the sum of the roots of x² − 8x + 15 = 0? (a) 7 (b) 8 (c) 9 (d) 15
  35. 35.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4925 (b) 5635 (c) 4225 (d) 5625
  36. 36.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1034 (b) 904 (c) 1024 (d) 784
  37. 37.What are the roots of x² − 11x + 28 = 0? (a) 2 and 14 (b) 1 and 10 (c) 1 and 28 (d) 4 and 7
  38. 38.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 12) (c) (x + 1)(x + 42) (d) (x + 6)(x + 7)
  39. 39.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4366 (b) 4884 (c) 4356 (d) 5476
  40. 40.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x − 3)(x − 8) (c) (x + 3)(x + 8) (d) (x + 1)(x + 24)

Answer key

  1. 7
  2. (x + 3)(x + 5)
  3. 4761
  4. 130
  5. 5 and 9
  6. 11025
  7. 25
  8. 48
  9. 12
  10. 10609
  11. 146
  12. 5
  13. 65
  14. 36
  15. (x + 8)(x + 9)
  16. 16
  17. 72
  18. 73
  19. 145
  20. (x + 7)(x + 8)
  21. 1936
  22. (x + 2)(x + 8)
  23. 3 and 7
  24. 16
  25. (x + 4)(x + 8)
  26. 5625
  27. 7 and 8
  28. 9
  29. 35
  30. 3 and 5
  31. 40
  32. (x + 2)(x + 7)
  33. 20
  34. 8
  35. 5625
  36. 1024
  37. 4 and 7
  38. (x + 6)(x + 7)
  39. 4356
  40. (x + 3)(x + 8)

Free maths olympiad lessons, practice and worksheets at talentjr.in/maths-olympiad