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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 766 · 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 − 17x + 72 = 0? (a) 72 (b) 17 (c) 18 (d) 16
  2. 2.What is the sum of the roots of x² − 9x + 18 = 0? (a) 9 (b) 8 (c) 18 (d) 10
  3. 3.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 15) (c) (x + 1)(x + 63) (d) (x + 7)(x + 9)
  4. 4.What is the product of the roots of x² − 8x + 12 = 0? (a) 8 (b) 12 (c) 11 (d) 13
  5. 5.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x + 1)(x + 20) (c) (x − 4)(x − 5) (d) (x + 1)(x + 8)
  6. 6.What are the roots of x² − 11x + 24 = 0? (a) 1 and 24 (b) 1 and 10 (c) 2 and 12 (d) 3 and 8
  7. 7.If a + b = 15 and ab = 56, what is (a − b)²? (a) 113 (b) 1 (c) 225 (d) 169
  8. 8.If a + b = 13 and ab = 30, what is a² + b²? (a) 169 (b) 109 (c) 229 (d) 139
  9. 9.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 72 (c) 73 (d) 17
  10. 10.If a + b = 19 and ab = 84, what is (a − b)²? (a) 193 (b) 361 (c) 25 (d) 5
  11. 11.If a + b = 17 and ab = 72, what is (a − b)²? (a) 217 (b) 145 (c) 1 (d) 289
  12. 12.If a + b = 15 and ab = 54, what is (a − b)²? (a) 117 (b) 225 (c) 9 (d) 3
  13. 13.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 54) (c) (x + 6)(x + 9) (d) (x + 1)(x + 14)
  14. 14.If a + b = 12 and ab = 35, what is a² + b²? (a) 144 (b) 214 (c) 109 (d) 74
  15. 15.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 7569 (b) 8091 (c) 8649 (d) 7579
  16. 16.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 8075 (b) 9025 (c) 7235 (d) 7225
  17. 17.If a + b = 18 and ab = 80, what is a² + b²? (a) 244 (b) 164 (c) 324 (d) 484
  18. 18.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1616 (b) 1936 (c) 1946 (d) 1296
  19. 19.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 16)
  20. 20.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 1)(x + 35) (c) (x + 1)(x + 11) (d) (x + 5)(x + 7)
  21. 21.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x − 2)(x − 6) (c) (x + 2)(x + 6) (d) (x + 1)(x + 12)
  22. 22.What are the roots of x² − 16x + 63 = 0? (a) 1 and 15 (b) 3 and 21 (c) 7 and 9 (d) 1 and 63
  23. 23.If a + b = 17 and ab = 66, what is (a − b)²? (a) 157 (b) 5 (c) 25 (d) 289
  24. 24.What is the sum of the roots of x² − 14x + 48 = 0? (a) 15 (b) 13 (c) 48 (d) 14
  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.If a + b = 9 and ab = 14, what is (a − b)²? (a) 81 (b) 5 (c) 25 (d) 53
  27. 27.What is the product of the roots of x² − 13x + 42 = 0? (a) 42 (b) 13 (c) 41 (d) 43
  28. 28.What is the sum of the roots of x² − 11x + 18 = 0? (a) 12 (b) 10 (c) 11 (d) 18
  29. 29.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x + 1)(x + 18) (c) (x + 2)(x + 9) (d) (x − 2)(x − 9)
  30. 30.What are the roots of x² − 8x + 15 = 0? (a) 2 and 6 (b) 1 and 15 (c) 1 and 7 (d) 3 and 5
  31. 31.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 9025 (b) 10025 (c) 11035 (d) 11025
  32. 32.If a + b = 12 and ab = 32, what is a² + b²? (a) 80 (b) 144 (c) 208 (d) 112
  33. 33.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 56 (c) 55 (d) 57
  34. 34.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 441 (c) 3 (d) 9
  35. 35.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7235 (b) 6425 (c) 5625 (d) 7225
  36. 36.If a + b = 10 and ab = 21, what is a² + b²? (a) 79 (b) 100 (c) 58 (d) 142
  37. 37.What is the sum of the roots of x² − 10x + 16 = 0? (a) 10 (b) 11 (c) 16 (d) 9
  38. 38.What is the sum of the roots of x² − 9x + 14 = 0? (a) 14 (b) 10 (c) 8 (d) 9
  39. 39.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 404 (b) 494 (c) 484 (d) 324
  40. 40.If a + b = 14 and ab = 45, what is a² + b²? (a) 151 (b) 196 (c) 106 (d) 286

Answer key

  1. 17
  2. 9
  3. (x + 7)(x + 9)
  4. 12
  5. (x + 4)(x + 5)
  6. 3 and 8
  7. 1
  8. 109
  9. 72
  10. 25
  11. 1
  12. 9
  13. (x + 6)(x + 9)
  14. 74
  15. 7569
  16. 7225
  17. 164
  18. 1936
  19. (x + 8)(x + 9)
  20. (x + 5)(x + 7)
  21. (x + 2)(x + 6)
  22. 7 and 9
  23. 25
  24. 14
  25. (x + 2)(x + 8)
  26. 25
  27. 42
  28. 11
  29. (x + 2)(x + 9)
  30. 3 and 5
  31. 11025
  32. 80
  33. 56
  34. 9
  35. 7225
  36. 58
  37. 10
  38. 9
  39. 484
  40. 106

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