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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 70 · 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 = 70, what is a² + b²? (a) 149 (b) 429 (c) 289 (d) 219
  2. 2.If a + b = 19 and ab = 84, what is a² + b²? (a) 529 (b) 361 (c) 277 (d) 193
  3. 3.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x + 1)(x + 18) (c) (x − 2)(x − 9) (d) (x + 2)(x + 9)
  4. 4.What is the product of the roots of x² − 11x + 24 = 0? (a) 25 (b) 24 (c) 11 (d) 23
  5. 5.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 1 and 72 (c) 8 and 9 (d) 2 and 36
  6. 6.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 8)(x + 9) (d) (x + 1)(x + 16)
  7. 7.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8474 (b) 7744 (c) 8464 (d) 8104
  8. 8.If a + b = 13 and ab = 42, what is (a − b)²? (a) 169 (b) 127 (c) 1 (d) 85
  9. 9.What is the product of the roots of x² − 13x + 36 = 0? (a) 13 (b) 35 (c) 36 (d) 37
  10. 10.If a + b = 12 and ab = 35, what is a² + b²? (a) 109 (b) 214 (c) 74 (d) 144
  11. 11.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1306 (b) 1296 (c) 1936 (d) 1584
  12. 12.If a + b = 8 and ab = 15, what is a² + b²? (a) 49 (b) 34 (c) 64 (d) 94
  13. 13.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10201 (b) 10211 (c) 9801 (d) 10001
  14. 14.If a + b = 5 and ab = 4, what is a² + b²? (a) 17 (b) 21 (c) 33 (d) 25
  15. 15.What is the sum of the roots of x² − 11x + 30 = 0? (a) 30 (b) 11 (c) 12 (d) 10
  16. 16.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x + 1)(x + 21) (c) (x + 3)(x + 7) (d) (x − 3)(x − 7)
  17. 17.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7056 (b) 7066 (c) 5776 (d) 6416
  18. 18.If a + b = 10 and ab = 16, what is a² + b²? (a) 84 (b) 68 (c) 132 (d) 100
  19. 19.If a + b = 18 and ab = 80, what is a² + b²? (a) 164 (b) 244 (c) 324 (d) 484
  20. 20.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 925 (b) 1235 (c) 625 (d) 1225
  21. 21.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 899 (b) 841 (c) 961 (d) 851
  22. 22.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 9409 (b) 10609 (c) 10009 (d) 10619
  23. 23.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 576 (b) 416 (c) 586 (d) 256
  24. 24.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6724 (b) 6396 (c) 6094 (d) 6084
  25. 25.If a + b = 18 and ab = 77, what is a² + b²? (a) 478 (b) 247 (c) 324 (d) 170
  26. 26.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 9025 (b) 11035 (c) 11025 (d) 10025
  27. 27.If a + b = 17 and ab = 70, what is (a − b)²? (a) 149 (b) 3 (c) 289 (d) 9
  28. 28.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 54) (c) (x + 6)(x + 9) (d) (x + 1)(x + 14)
  29. 29.If a + b = 17 and ab = 66, what is a² + b²? (a) 157 (b) 289 (c) 223 (d) 421
  30. 30.What is the product of the roots of x² − 10x + 21 = 0? (a) 10 (b) 20 (c) 22 (d) 21
  31. 31.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1946 (b) 1936 (c) 1296 (d) 1616
  32. 32.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3609 (b) 3249 (c) 3979 (d) 3969
  33. 33.What are the roots of x² − 11x + 24 = 0? (a) 1 and 24 (b) 1 and 10 (c) 2 and 12 (d) 3 and 8
  34. 34.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 2)(x + 4) (c) (x + 1)(x + 5) (d) (x + 1)(x + 8)
  35. 35.If a + b = 15 and ab = 36, what is (a − b)²? (a) 225 (b) 153 (c) 9 (d) 81
  36. 36.If a + b = 14 and ab = 45, what is (a − b)²? (a) 196 (b) 16 (c) 4 (d) 106
  37. 37.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9975 (b) 11025 (c) 9025 (d) 9035
  38. 38.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 451 (b) 441 (c) 401 (d) 361
  39. 39.What is the product of the roots of x² − 5x + 6 = 0? (a) 5 (b) 6 (c) 7 (d) 8
  40. 40.If a + b = 19 and ab = 84, what is (a − b)²? (a) 361 (b) 5 (c) 193 (d) 25

Answer key

  1. 149
  2. 193
  3. (x + 2)(x + 9)
  4. 24
  5. 8 and 9
  6. (x + 8)(x + 9)
  7. 8464
  8. 1
  9. 36
  10. 74
  11. 1296
  12. 34
  13. 10201
  14. 17
  15. 11
  16. (x + 3)(x + 7)
  17. 7056
  18. 68
  19. 164
  20. 1225
  21. 841
  22. 10609
  23. 576
  24. 6084
  25. 170
  26. 11025
  27. 9
  28. (x + 6)(x + 9)
  29. 157
  30. 21
  31. 1936
  32. 3969
  33. 3 and 8
  34. (x + 2)(x + 4)
  35. 81
  36. 16
  37. 9025
  38. 441
  39. 6
  40. 25

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