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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 707 · 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 = 8 and ab = 7, what is (a − b)²? (a) 36 (b) 64 (c) 6 (d) 50
  2. 2.What is the sum of the roots of x² − 11x + 28 = 0? (a) 10 (b) 28 (c) 12 (d) 11
  3. 3.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 484 (b) 334 (c) 396 (d) 324
  4. 4.If a + b = 18 and ab = 80, what is (a − b)²? (a) 164 (b) 324 (c) 4 (d) 2
  5. 5.What is the product of the roots of x² − 8x + 12 = 0? (a) 11 (b) 8 (c) 13 (d) 12
  6. 6.If a + b = 9 and ab = 20, what is (a − b)²? (a) 41 (b) 61 (c) 81 (d) 1
  7. 7.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8291 (b) 8101 (c) 8281 (d) 7921
  8. 8.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x − 8)(x − 9) (c) (x + 1)(x + 72) (d) (x + 1)(x + 16)
  9. 9.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3596 (b) 3844 (c) 3374 (d) 3364
  10. 10.If a + b = 10 and ab = 24, what is (a − b)²? (a) 4 (b) 52 (c) 2 (d) 100
  11. 11.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 909 (b) 1089 (c) 1099 (d) 729
  12. 12.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3616 (b) 4106 (c) 4096 (d) 3136
  13. 13.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x + 1)(x + 13) (c) (x + 1)(x + 45) (d) (x − 5)(x − 9)
  14. 14.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 891 (b) 739 (c) 1089 (d) 729
  15. 15.If a + b = 18 and ab = 72, what is a² + b²? (a) 468 (b) 252 (c) 180 (d) 324
  16. 16.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 63 (c) 16 (d) 17
  17. 17.What is the product of the roots of x² − 6x + 8 = 0? (a) 8 (b) 7 (c) 9 (d) 6
  18. 18.If a + b = 20 and ab = 96, what is a² + b²? (a) 304 (b) 208 (c) 592 (d) 400
  19. 19.What is the sum of the roots of x² − 11x + 18 = 0? (a) 10 (b) 12 (c) 11 (d) 18
  20. 20.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x − 5)(x − 7) (c) (x + 5)(x + 7) (d) (x + 1)(x + 35)
  21. 21.If a + b = 17 and ab = 72, what is a² + b²? (a) 217 (b) 433 (c) 289 (d) 145
  22. 22.If a + b = 17 and ab = 60, what is (a − b)²? (a) 289 (b) 169 (c) 49 (d) 7
  23. 23.What is the sum of the roots of x² − 9x + 20 = 0? (a) 8 (b) 20 (c) 10 (d) 9
  24. 24.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8846 (b) 8116 (c) 8836 (d) 7396
  25. 25.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 1)(x + 10) (c) (x + 1)(x + 28) (d) (x + 4)(x + 7)
  26. 26.What is the sum of the roots of x² − 5x + 6 = 0? (a) 6 (b) 5 (c) 7 (d) 4
  27. 27.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 299 (b) 529 (c) 289 (d) 391
  28. 28.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)
  29. 29.Which is the correct factorisation of x² + 6x + 8? (a) (x + 2)(x + 4) (b) (x + 1)(x + 8) (c) (x − 2)(x − 4) (d) (x + 1)(x + 5)
  30. 30.If a + b = 8 and ab = 15, what is a² + b²? (a) 34 (b) 64 (c) 94 (d) 49
  31. 31.If a + b = 12 and ab = 35, what is a² + b²? (a) 109 (b) 144 (c) 74 (d) 214
  32. 32.If a + b = 13 and ab = 36, what is (a − b)²? (a) 169 (b) 97 (c) 25 (d) 5
  33. 33.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 7931 (b) 7921 (c) 8281 (d) 8099
  34. 34.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1454 (b) 1444 (c) 1764 (d) 1596
  35. 35.If a + b = 14 and ab = 45, what is (a − b)²? (a) 106 (b) 4 (c) 196 (d) 16
  36. 36.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x + 3)(x + 4) (c) (x + 1)(x + 6) (d) (x − 3)(x − 4)
  37. 37.If a + b = 18 and ab = 80, what is a² + b²? (a) 244 (b) 324 (c) 164 (d) 484
  38. 38.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 11025 (b) 9035 (c) 9975 (d) 9025
  39. 39.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1156 (b) 676 (c) 1166 (d) 916
  40. 40.What is the sum of the roots of x² − 7x + 10 = 0? (a) 10 (b) 8 (c) 6 (d) 7

Answer key

  1. 36
  2. 11
  3. 324
  4. 4
  5. 12
  6. 1
  7. 8281
  8. (x + 8)(x + 9)
  9. 3364
  10. 4
  11. 1089
  12. 4096
  13. (x + 5)(x + 9)
  14. 729
  15. 180
  16. 16
  17. 8
  18. 208
  19. 11
  20. (x + 5)(x + 7)
  21. 145
  22. 49
  23. 9
  24. 8836
  25. (x + 4)(x + 7)
  26. 5
  27. 289
  28. (x + 6)(x + 9)
  29. (x + 2)(x + 4)
  30. 34
  31. 74
  32. 25
  33. 7921
  34. 1444
  35. 16
  36. (x + 3)(x + 4)
  37. 164
  38. 9025
  39. 1156
  40. 7

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