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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 59 · 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 = 17 and ab = 72, what is a² + b²? (a) 217 (b) 145 (c) 289 (d) 433
  2. 2.If a + b = 14 and ab = 48, what is (a − b)²? (a) 2 (b) 100 (c) 4 (d) 196
  3. 3.What is the sum of the roots of x² − 12x + 27 = 0? (a) 12 (b) 11 (c) 27 (d) 13
  4. 4.What are the roots of x² − 8x + 12 = 0? (a) 1 and 7 (b) 3 and 4 (c) 1 and 12 (d) 2 and 6
  5. 5.What is the product of the roots of x² − 14x + 45 = 0? (a) 14 (b) 46 (c) 45 (d) 44
  6. 6.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3609 (b) 3249 (c) 3969 (d) 3979
  7. 7.What are the roots of x² − 13x + 42 = 0? (a) 1 and 12 (b) 1 and 42 (c) 6 and 7 (d) 2 and 21
  8. 8.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1156 (b) 916 (c) 676 (d) 1166
  9. 9.If a + b = 3 and ab = 2, what is a² + b²? (a) 7 (b) 9 (c) 5 (d) 13
  10. 10.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6724 (b) 6084 (c) 6734 (d) 6404
  11. 11.If a + b = 15 and ab = 36, what is a² + b²? (a) 225 (b) 189 (c) 153 (d) 297
  12. 12.What is the sum of the roots of x² − 7x + 10 = 0? (a) 10 (b) 8 (c) 6 (d) 7
  13. 13.What is the product of the roots of x² − 12x + 27 = 0? (a) 27 (b) 12 (c) 26 (d) 28
  14. 14.What is the sum of the roots of x² − 12x + 32 = 0? (a) 13 (b) 11 (c) 32 (d) 12
  15. 15.Which is the correct factorisation of x² + 11x + 28? (a) (x + 4)(x + 7) (b) (x + 1)(x + 28) (c) (x + 1)(x + 10) (d) (x − 4)(x − 7)
  16. 16.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 9) (b) (x − 4)(x − 6) (c) (x + 1)(x + 24) (d) (x + 4)(x + 6)
  17. 17.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 404 (b) 484 (c) 324 (d) 494
  18. 18.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x − 2)(x − 7) (c) (x + 1)(x + 8) (d) (x + 2)(x + 7)
  19. 19.What is the sum of the roots of x² − 11x + 24 = 0? (a) 10 (b) 12 (c) 24 (d) 11
  20. 20.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x − 5)(x − 9) (c) (x + 5)(x + 9) (d) (x + 1)(x + 45)
  21. 21.If a + b = 15 and ab = 54, what is (a − b)²? (a) 9 (b) 117 (c) 3 (d) 225
  22. 22.If a + b = 17 and ab = 70, what is a² + b²? (a) 219 (b) 429 (c) 149 (d) 289
  23. 23.What is the product of the roots of x² − 10x + 24 = 0? (a) 24 (b) 25 (c) 10 (d) 23
  24. 24.If a + b = 12 and ab = 27, what is (a − b)²? (a) 36 (b) 144 (c) 90 (d) 6
  25. 25.What is the product of the roots of x² − 15x + 54 = 0? (a) 53 (b) 54 (c) 55 (d) 15
  26. 26.If a + b = 14 and ab = 48, what is a² + b²? (a) 292 (b) 148 (c) 196 (d) 100
  27. 27.What are the roots of x² − 15x + 54 = 0? (a) 1 and 14 (b) 6 and 9 (c) 2 and 27 (d) 1 and 54
  28. 28.What is the sum of the roots of x² − 13x + 40 = 0? (a) 40 (b) 14 (c) 13 (d) 12
  29. 29.If a + b = 21 and ab = 108, what is a² + b²? (a) 225 (b) 657 (c) 441 (d) 333
  30. 30.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1454 (b) 1444 (c) 1764 (d) 1596
  31. 31.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2916 (b) 2116 (c) 2516 (d) 2926
  32. 32.If a + b = 13 and ab = 40, what is a² + b²? (a) 89 (b) 249 (c) 169 (d) 129
  33. 33.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1225 (b) 1575 (c) 1235 (d) 2025
  34. 34.If a + b = 14 and ab = 45, what is (a − b)²? (a) 4 (b) 16 (c) 196 (d) 106
  35. 35.If a + b = 15 and ab = 56, what is (a − b)²? (a) 1 (b) 113 (c) 225 (d) 169
  36. 36.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 5041 (b) 4771 (c) 4899 (d) 4761
  37. 37.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 9604 (b) 10414 (c) 10004 (d) 10404
  38. 38.What are the roots of x² − 15x + 56 = 0? (a) 1 and 14 (b) 1 and 56 (c) 2 and 28 (d) 7 and 8
  39. 39.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 401 (b) 441 (c) 361 (d) 451
  40. 40.If a + b = 11 and ab = 28, what is a² + b²? (a) 65 (b) 93 (c) 121 (d) 177

Answer key

  1. 145
  2. 4
  3. 12
  4. 2 and 6
  5. 45
  6. 3969
  7. 6 and 7
  8. 1156
  9. 5
  10. 6724
  11. 153
  12. 7
  13. 27
  14. 12
  15. (x + 4)(x + 7)
  16. (x + 4)(x + 6)
  17. 484
  18. (x + 2)(x + 7)
  19. 11
  20. (x + 5)(x + 9)
  21. 9
  22. 149
  23. 24
  24. 36
  25. 54
  26. 100
  27. 6 and 9
  28. 13
  29. 225
  30. 1444
  31. 2916
  32. 89
  33. 1225
  34. 16
  35. 1
  36. 4761
  37. 10404
  38. 7 and 8
  39. 441
  40. 65

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