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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 638 · 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 product of the roots of x² − 11x + 18 = 0? (a) 18 (b) 11 (c) 17 (d) 19
  2. 2.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x + 1)(x + 20) (c) (x + 4)(x + 5) (d) (x − 4)(x − 5)
  3. 3.If a + b = 14 and ab = 24, what is a² + b²? (a) 172 (b) 244 (c) 196 (d) 148
  4. 4.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1681 (b) 1691 (c) 1601 (d) 1521
  5. 5.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 2 and 36 (c) 8 and 9 (d) 1 and 72
  6. 6.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 14) (c) (x + 7)(x + 8) (d) (x + 1)(x + 56)
  7. 7.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x − 2)(x − 8) (c) (x + 2)(x + 8) (d) (x + 1)(x + 16)
  8. 8.If a + b = 15 and ab = 56, what is a² + b²? (a) 113 (b) 169 (c) 337 (d) 225
  9. 9.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
  10. 10.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x + 2)(x + 9) (c) (x − 2)(x − 9) (d) (x + 1)(x + 18)
  11. 11.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x + 1)(x + 35) (c) (x − 5)(x − 7) (d) (x + 5)(x + 7)
  12. 12.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 416 (c) 586 (d) 576
  13. 13.If a + b = 11 and ab = 28, what is a² + b²? (a) 121 (b) 93 (c) 177 (d) 65
  14. 14.If a + b = 17 and ab = 70, what is a² + b²? (a) 289 (b) 219 (c) 429 (d) 149
  15. 15.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)
  16. 16.If a + b = 18 and ab = 80, what is a² + b²? (a) 484 (b) 324 (c) 164 (d) 244
  17. 17.If a + b = 13 and ab = 36, what is a² + b²? (a) 241 (b) 169 (c) 133 (d) 97
  18. 18.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x + 3)(x + 6) (c) (x − 3)(x − 6) (d) (x + 1)(x + 8)
  19. 19.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 5)(x + 6) (c) (x − 5)(x − 6) (d) (x + 1)(x + 10)
  20. 20.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x + 5)(x + 9) (c) (x + 1)(x + 45) (d) (x − 5)(x − 9)
  21. 21.If a + b = 14 and ab = 40, what is a² + b²? (a) 196 (b) 276 (c) 156 (d) 116
  22. 22.If a + b = 3 and ab = 2, what is a² + b²? (a) 13 (b) 5 (c) 9 (d) 7
  23. 23.Which is the correct factorisation of x² + 11x + 24? (a) (x + 3)(x + 8) (b) (x + 1)(x + 10) (c) (x + 1)(x + 24) (d) (x − 3)(x − 8)
  24. 24.What are the roots of x² − 16x + 63 = 0? (a) 1 and 15 (b) 1 and 63 (c) 7 and 9 (d) 3 and 21
  25. 25.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9984 (b) 10816 (c) 9226 (d) 9216
  26. 26.If a + b = 9 and ab = 20, what is a² + b²? (a) 61 (b) 121 (c) 81 (d) 41
  27. 27.If a + b = 13 and ab = 40, what is (a − b)²? (a) 89 (b) 169 (c) 3 (d) 9
  28. 28.What is the sum of the roots of x² − 15x + 54 = 0? (a) 16 (b) 54 (c) 15 (d) 14
  29. 29.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 676 (b) 916 (c) 1156 (d) 1166
  30. 30.If a + b = 13 and ab = 12, what is a² + b²? (a) 193 (b) 157 (c) 169 (d) 145
  31. 31.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 73 (c) 72 (d) 17
  32. 32.If a + b = 17 and ab = 72, what is a² + b²? (a) 145 (b) 433 (c) 217 (d) 289
  33. 33.What is the sum of the roots of x² − 8x + 12 = 0? (a) 9 (b) 8 (c) 7 (d) 12
  34. 34.If a + b = 13 and ab = 22, what is (a − b)²? (a) 81 (b) 169 (c) 9 (d) 125
  35. 35.What are the roots of x² − 14x + 48 = 0? (a) 2 and 24 (b) 6 and 8 (c) 1 and 13 (d) 1 and 48
  36. 36.If a + b = 10 and ab = 24, what is a² + b²? (a) 148 (b) 76 (c) 100 (d) 52
  37. 37.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x − 6)(x − 7) (c) (x + 1)(x + 42) (d) (x + 1)(x + 12)
  38. 38.If a + b = 14 and ab = 45, what is a² + b²? (a) 151 (b) 286 (c) 106 (d) 196
  39. 39.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 18 (c) 72 (d) 17
  40. 40.If a + b = 21 and ab = 108, what is (a − b)²? (a) 3 (b) 441 (c) 225 (d) 9

Answer key

  1. 18
  2. (x + 4)(x + 5)
  3. 148
  4. 1681
  5. 8 and 9
  6. (x + 7)(x + 8)
  7. (x + 2)(x + 8)
  8. 113
  9. (x + 8)(x + 9)
  10. (x + 2)(x + 9)
  11. (x + 5)(x + 7)
  12. 576
  13. 65
  14. 149
  15. (x + 7)(x + 9)
  16. 164
  17. 97
  18. (x + 3)(x + 6)
  19. (x + 5)(x + 6)
  20. (x + 5)(x + 9)
  21. 116
  22. 5
  23. (x + 3)(x + 8)
  24. 7 and 9
  25. 9216
  26. 41
  27. 9
  28. 15
  29. 1156
  30. 145
  31. 72
  32. 145
  33. 8
  34. 81
  35. 6 and 8
  36. 52
  37. (x + 6)(x + 7)
  38. 106
  39. 17
  40. 9

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