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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 377 · 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.If a + b = 7 and ab = 12, what is (a − b)²? (a) 25 (b) 49 (c) 1 (d) 37
  2. 2.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 63) (c) (x + 1)(x + 15) (d) (x + 7)(x + 9)
  3. 3.If a + b = 16 and ab = 63, what is (a − b)²? (a) 256 (b) 4 (c) 130 (d) 2
  4. 4.What is the product of the roots of x² − 7x + 10 = 0? (a) 9 (b) 7 (c) 10 (d) 11
  5. 5.If a + b = 5 and ab = 4, what is a² + b²? (a) 17 (b) 21 (c) 25 (d) 33
  6. 6.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 5)(x + 6) (c) (x + 1)(x + 30) (d) (x − 5)(x − 6)
  7. 7.If a + b = 15 and ab = 54, what is (a − b)²? (a) 117 (b) 9 (c) 3 (d) 225
  8. 8.If a + b = 6 and ab = 5, what is (a − b)²? (a) 26 (b) 16 (c) 36 (d) 4
  9. 9.If a + b = 15 and ab = 36, what is a² + b²? (a) 297 (b) 153 (c) 225 (d) 189
  10. 10.If a + b = 11 and ab = 28, what is a² + b²? (a) 121 (b) 177 (c) 65 (d) 93
  11. 11.If a + b = 18 and ab = 72, what is (a − b)²? (a) 36 (b) 324 (c) 180 (d) 6
  12. 12.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 16 (d) 17
  13. 13.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1521 (b) 1531 (c) 1681 (d) 1599
  14. 14.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x + 1)(x + 8) (c) (x − 2)(x − 4) (d) (x + 2)(x + 4)
  15. 15.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 289 (c) 145 (d) 217
  16. 16.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)
  17. 17.If a + b = 14 and ab = 45, what is (a − b)²? (a) 106 (b) 16 (c) 196 (d) 4
  18. 18.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x − 3)(x − 5) (c) (x + 3)(x + 5) (d) (x + 1)(x + 7)
  19. 19.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 32) (c) (x + 1)(x + 11) (d) (x + 4)(x + 8)
  20. 20.If a + b = 11 and ab = 18, what is (a − b)²? (a) 85 (b) 121 (c) 7 (d) 49
  21. 21.If a + b = 13 and ab = 12, what is a² + b²? (a) 193 (b) 157 (c) 169 (d) 145
  22. 22.What is the product of the roots of x² − 12x + 32 = 0? (a) 12 (b) 31 (c) 32 (d) 33
  23. 23.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 225 (c) 657 (d) 441
  24. 24.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x − 6)(x − 9) (c) (x + 1)(x + 14) (d) (x + 1)(x + 54)
  25. 25.If a + b = 14 and ab = 48, what is (a − b)²? (a) 100 (b) 4 (c) 196 (d) 2
  26. 26.If a + b = 13 and ab = 22, what is (a − b)²? (a) 81 (b) 9 (c) 125 (d) 169
  27. 27.What is the product of the roots of x² − 10x + 21 = 0? (a) 21 (b) 20 (c) 10 (d) 22
  28. 28.What is the sum of the roots of x² − 7x + 10 = 0? (a) 6 (b) 10 (c) 8 (d) 7
  29. 29.If a + b = 20 and ab = 99, what is (a − b)²? (a) 202 (b) 4 (c) 2 (d) 400
  30. 30.Which is the correct factorisation of x² + 8x + 12? (a) (x − 2)(x − 6) (b) (x + 2)(x + 6) (c) (x + 1)(x + 12) (d) (x + 1)(x + 7)
  31. 31.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 9025 (b) 8075 (c) 7235 (d) 7225
  32. 32.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1296 (b) 1306 (c) 1584 (d) 1936
  33. 33.What is the product of the roots of x² − 10x + 24 = 0? (a) 25 (b) 23 (c) 24 (d) 10
  34. 34.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)
  35. 35.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 8836 (b) 7396 (c) 8084 (d) 7406
  36. 36.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 73 (c) 71 (d) 72
  37. 37.What is the sum of the roots of x² − 8x + 12 = 0? (a) 9 (b) 12 (c) 8 (d) 7
  38. 38.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x − 3)(x − 7) (c) (x + 3)(x + 7) (d) (x + 1)(x + 9)
  39. 39.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 12) (c) (x + 6)(x + 7) (d) (x + 1)(x + 42)
  40. 40.If a + b = 12 and ab = 32, what is (a − b)²? (a) 144 (b) 80 (c) 16 (d) 4

Answer key

  1. 1
  2. (x + 7)(x + 9)
  3. 4
  4. 10
  5. 17
  6. (x + 5)(x + 6)
  7. 9
  8. 16
  9. 153
  10. 65
  11. 36
  12. 17
  13. 1521
  14. (x + 2)(x + 4)
  15. 145
  16. (x + 7)(x + 8)
  17. 16
  18. (x + 3)(x + 5)
  19. (x + 4)(x + 8)
  20. 49
  21. 145
  22. 32
  23. 225
  24. (x + 6)(x + 9)
  25. 4
  26. 81
  27. 21
  28. 7
  29. 4
  30. (x + 2)(x + 6)
  31. 7225
  32. 1296
  33. 24
  34. (x + 8)(x + 9)
  35. 7396
  36. 72
  37. 8
  38. (x + 3)(x + 7)
  39. (x + 6)(x + 7)
  40. 16

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