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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 535 · 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 = 16 and ab = 55, what is a² + b²? (a) 256 (b) 366 (c) 146 (d) 201
  2. 2.If a + b = 16 and ab = 60, what is (a − b)²? (a) 4 (b) 256 (c) 16 (d) 136
  3. 3.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)
  4. 4.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x + 4)(x + 9) (c) (x − 4)(x − 9) (d) (x + 1)(x + 36)
  5. 5.What are the roots of x² − 15x + 56 = 0? (a) 1 and 14 (b) 2 and 28 (c) 1 and 56 (d) 7 and 8
  6. 6.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
  7. 7.If a + b = 18 and ab = 77, what is a² + b²? (a) 324 (b) 170 (c) 478 (d) 247
  8. 8.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10004 (b) 9604 (c) 10414 (d) 10404
  9. 9.If a + b = 12 and ab = 20, what is a² + b²? (a) 124 (b) 184 (c) 144 (d) 104
  10. 10.What is the product of the roots of x² − 14x + 48 = 0? (a) 48 (b) 14 (c) 47 (d) 49
  11. 11.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x − 7)(x − 8) (d) (x + 7)(x + 8)
  12. 12.What is the sum of the roots of x² − 11x + 18 = 0? (a) 10 (b) 18 (c) 12 (d) 11
  13. 13.If a + b = 17 and ab = 70, what is a² + b²? (a) 219 (b) 429 (c) 289 (d) 149
  14. 14.What are the roots of x² − 8x + 12 = 0? (a) 1 and 12 (b) 1 and 7 (c) 2 and 6 (d) 3 and 4
  15. 15.If a + b = 10 and ab = 21, what is a² + b²? (a) 58 (b) 79 (c) 142 (d) 100
  16. 16.If a + b = 14 and ab = 24, what is a² + b²? (a) 196 (b) 148 (c) 244 (d) 172
  17. 17.What is the sum of the roots of x² − 5x + 6 = 0? (a) 6 (b) 4 (c) 7 (d) 5
  18. 18.If a + b = 10 and ab = 21, what is (a − b)²? (a) 4 (b) 58 (c) 16 (d) 100
  19. 19.What is the sum of the roots of x² − 8x + 12 = 0? (a) 9 (b) 8 (c) 12 (d) 7
  20. 20.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x − 5)(x − 8) (c) (x + 5)(x + 8) (d) (x + 1)(x + 12)
  21. 21.If a + b = 4 and ab = 3, what is (a − b)²? (a) 4 (b) 2 (c) 10 (d) 16
  22. 22.If a + b = 7 and ab = 6, what is a² + b²? (a) 61 (b) 43 (c) 37 (d) 49
  23. 23.Which is the correct factorisation of x² + 8x + 15? (a) (x + 3)(x + 5) (b) (x − 3)(x − 5) (c) (x + 1)(x + 15) (d) (x + 1)(x + 7)
  24. 24.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 6416 (b) 7066 (c) 5776 (d) 7056
  25. 25.If a + b = 9 and ab = 20, what is a² + b²? (a) 81 (b) 61 (c) 121 (d) 41
  26. 26.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 494 (b) 404 (c) 324 (d) 484
  27. 27.What is the sum of the roots of x² − 10x + 21 = 0? (a) 21 (b) 11 (c) 10 (d) 9
  28. 28.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2819 (b) 2509 (c) 2809 (d) 2209
  29. 29.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1099 (b) 909 (c) 729 (d) 1089
  30. 30.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2926 (b) 2516 (c) 2116 (d) 2916
  31. 31.What is the sum of the roots of x² − 10x + 16 = 0? (a) 9 (b) 16 (c) 10 (d) 11
  32. 32.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9984 (b) 9226 (c) 9216 (d) 10816
  33. 33.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6401 (b) 6241 (c) 6561 (d) 6571
  34. 34.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 16 (d) 18
  35. 35.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1235 (b) 1575 (c) 1225 (d) 2025
  36. 36.If a + b = 14 and ab = 45, what is a² + b²? (a) 286 (b) 106 (c) 151 (d) 196
  37. 37.If a + b = 13 and ab = 30, what is a² + b²? (a) 229 (b) 169 (c) 109 (d) 139
  38. 38.If a + b = 14 and ab = 33, what is a² + b²? (a) 262 (b) 130 (c) 196 (d) 163
  39. 39.What is the product of the roots of x² − 11x + 24 = 0? (a) 23 (b) 25 (c) 24 (d) 11
  40. 40.If a + b = 18 and ab = 80, what is a² + b²? (a) 244 (b) 324 (c) 484 (d) 164

Answer key

  1. 146
  2. 16
  3. (x + 7)(x + 9)
  4. (x + 4)(x + 9)
  5. 7 and 8
  6. (x + 8)(x + 9)
  7. 170
  8. 10404
  9. 104
  10. 48
  11. (x + 7)(x + 8)
  12. 11
  13. 149
  14. 2 and 6
  15. 58
  16. 148
  17. 5
  18. 16
  19. 8
  20. (x + 5)(x + 8)
  21. 4
  22. 37
  23. (x + 3)(x + 5)
  24. 7056
  25. 41
  26. 484
  27. 10
  28. 2809
  29. 1089
  30. 2916
  31. 10
  32. 9216
  33. 6561
  34. 17
  35. 1225
  36. 106
  37. 109
  38. 130
  39. 24
  40. 164

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