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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 826 · 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 − 12x + 27 = 0? (a) 12 (b) 27 (c) 13 (d) 11
  2. 2.If a + b = 18 and ab = 77, what is a² + b²? (a) 247 (b) 478 (c) 324 (d) 170
  3. 3.What is the product of the roots of x² − 11x + 24 = 0? (a) 11 (b) 23 (c) 24 (d) 25
  4. 4.If a + b = 17 and ab = 70, what is a² + b²? (a) 289 (b) 219 (c) 149 (d) 429
  5. 5.What are the roots of x² − 11x + 24 = 0? (a) 1 and 10 (b) 1 and 24 (c) 2 and 12 (d) 3 and 8
  6. 6.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 841 (b) 961 (c) 899 (d) 851
  7. 7.What are the roots of x² − 11x + 30 = 0? (a) 1 and 10 (b) 1 and 30 (c) 5 and 6 (d) 2 and 15
  8. 8.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 12) (c) (x + 1)(x + 6) (d) (x + 3)(x + 4)
  9. 9.If a + b = 5 and ab = 6, what is a² + b²? (a) 19 (b) 13 (c) 37 (d) 25
  10. 10.If a + b = 14 and ab = 48, what is a² + b²? (a) 196 (b) 100 (c) 148 (d) 292
  11. 11.If a + b = 15 and ab = 50, what is (a − b)²? (a) 225 (b) 5 (c) 25 (d) 125
  12. 12.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 5)(x + 6) (c) (x + 1)(x + 10) (d) (x − 5)(x − 6)
  13. 13.What are the roots of x² − 12x + 27 = 0? (a) 1 and 27 (b) 3 and 9 (c) 2 and 10 (d) 1 and 11
  14. 14.If a + b = 13 and ab = 30, what is a² + b²? (a) 139 (b) 169 (c) 229 (d) 109
  15. 15.If a + b = 5 and ab = 6, what is (a − b)²? (a) 13 (b) 25 (c) 19 (d) 1
  16. 16.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 1)(x + 15) (c) (x − 7)(x − 9) (d) (x + 7)(x + 9)
  17. 17.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 586 (c) 416 (d) 576
  18. 18.If a + b = 11 and ab = 30, what is a² + b²? (a) 181 (b) 121 (c) 91 (d) 61
  19. 19.If a + b = 10 and ab = 21, what is (a − b)²? (a) 4 (b) 16 (c) 100 (d) 58
  20. 20.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x − 7)(x − 8) (c) (x + 1)(x + 14) (d) (x + 7)(x + 8)
  21. 21.If a + b = 9 and ab = 8, what is a² + b²? (a) 65 (b) 97 (c) 73 (d) 81
  22. 22.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4225 (b) 5625 (c) 4235 (d) 4875
  23. 23.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6571 (b) 6561 (c) 6241 (d) 6401
  24. 24.If a + b = 3 and ab = 2, what is (a − b)²? (a) 7 (b) 1 (c) 9 (d) 5
  25. 25.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6399 (b) 6251 (c) 6561 (d) 6241
  26. 26.If a + b = 20 and ab = 96, what is a² + b²? (a) 208 (b) 400 (c) 304 (d) 592
  27. 27.If a + b = 9 and ab = 14, what is a² + b²? (a) 81 (b) 109 (c) 53 (d) 67
  28. 28.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 145 (c) 289 (d) 217
  29. 29.If a + b = 9 and ab = 14, what is (a − b)²? (a) 25 (b) 81 (c) 53 (d) 5
  30. 30.If a + b = 21 and ab = 108, what is (a − b)²? (a) 9 (b) 441 (c) 3 (d) 225
  31. 31.What is the sum of the roots of x² − 6x + 8 = 0? (a) 8 (b) 6 (c) 7 (d) 5
  32. 32.What is the product of the roots of x² − 12x + 27 = 0? (a) 26 (b) 28 (c) 12 (d) 27
  33. 33.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 10816 (b) 9216 (c) 9984 (d) 9226
  34. 34.If a + b = 12 and ab = 20, what is a² + b²? (a) 144 (b) 104 (c) 184 (d) 124
  35. 35.If a + b = 17 and ab = 60, what is a² + b²? (a) 409 (b) 289 (c) 229 (d) 169
  36. 36.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 8 and 9 (c) 1 and 16 (d) 2 and 36
  37. 37.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1099 (b) 909 (c) 729 (d) 1089
  38. 38.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 11) (b) (x − 4)(x − 8) (c) (x + 4)(x + 8) (d) (x + 1)(x + 32)
  39. 39.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 9025 (b) 11035 (c) 10025 (d) 11025
  40. 40.What is the product of the roots of x² − 14x + 48 = 0? (a) 48 (b) 47 (c) 49 (d) 14

Answer key

  1. 12
  2. 170
  3. 24
  4. 149
  5. 3 and 8
  6. 841
  7. 5 and 6
  8. (x + 3)(x + 4)
  9. 13
  10. 100
  11. 25
  12. (x + 5)(x + 6)
  13. 3 and 9
  14. 109
  15. 1
  16. (x + 7)(x + 9)
  17. 576
  18. 61
  19. 16
  20. (x + 7)(x + 8)
  21. 65
  22. 4225
  23. 6561
  24. 1
  25. 6241
  26. 208
  27. 53
  28. 145
  29. 25
  30. 9
  31. 6
  32. 27
  33. 9216
  34. 104
  35. 169
  36. 8 and 9
  37. 1089
  38. (x + 4)(x + 8)
  39. 11025
  40. 48

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