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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 506 · 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.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 1)(x + 10) (c) (x − 5)(x − 6) (d) (x + 5)(x + 6)
  2. 2.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8291 (b) 8281 (c) 7921 (d) 8101
  3. 3.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 15 (c) 56 (d) 57
  4. 4.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6889 (b) 5929 (c) 5939 (d) 6391
  5. 5.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1235 (b) 925 (c) 625 (d) 1225
  6. 6.What is the sum of the roots of x² − 13x + 36 = 0? (a) 13 (b) 36 (c) 12 (d) 14
  7. 7.What is the product of the roots of x² − 13x + 42 = 0? (a) 13 (b) 42 (c) 41 (d) 43
  8. 8.What is the product of the roots of x² − 12x + 27 = 0? (a) 26 (b) 28 (c) 12 (d) 27
  9. 9.If a + b = 11 and ab = 18, what is a² + b²? (a) 157 (b) 121 (c) 103 (d) 85
  10. 10.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 909 (b) 729 (c) 1099 (d) 1089
  11. 11.What is the product of the roots of x² − 7x + 10 = 0? (a) 11 (b) 9 (c) 10 (d) 7
  12. 12.If a + b = 9 and ab = 18, what is (a − b)²? (a) 3 (b) 45 (c) 81 (d) 9
  13. 13.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 5625 (b) 4875 (c) 4225 (d) 4235
  14. 14.If a + b = 13 and ab = 22, what is (a − b)²? (a) 125 (b) 169 (c) 81 (d) 9
  15. 15.What are the roots of x² − 9x + 20 = 0? (a) 4 and 5 (b) 1 and 20 (c) 2 and 10 (d) 1 and 8
  16. 16.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 884 (b) 686 (c) 1156 (d) 676
  17. 17.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
  18. 18.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9975 (b) 9035 (c) 9025 (d) 11025
  19. 19.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 16 (c) 17 (d) 15
  20. 20.If a + b = 10 and ab = 21, what is (a − b)²? (a) 100 (b) 16 (c) 58 (d) 4
  21. 21.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 72 (c) 16 (d) 17
  22. 22.What is the sum of the roots of x² − 7x + 12 = 0? (a) 6 (b) 8 (c) 7 (d) 12
  23. 23.If a + b = 12 and ab = 11, what is (a − b)²? (a) 100 (b) 144 (c) 122 (d) 10
  24. 24.Which is the correct factorisation of x² + 12x + 32? (a) (x + 4)(x + 8) (b) (x + 1)(x + 32) (c) (x − 4)(x − 8) (d) (x + 1)(x + 11)
  25. 25.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 181 (c) 361 (d) 271
  26. 26.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 8 and 9 (c) 1 and 72 (d) 1 and 16
  27. 27.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3374 (b) 3364 (c) 3844 (d) 3596
  28. 28.If a + b = 4 and ab = 3, what is a² + b²? (a) 22 (b) 10 (c) 16 (d) 13
  29. 29.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 42) (c) (x + 1)(x + 12) (d) (x + 6)(x + 7)
  30. 30.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x + 1)(x + 8) (c) (x + 4)(x + 5) (d) (x − 4)(x − 5)
  31. 31.What is the product of the roots of x² − 8x + 12 = 0? (a) 8 (b) 12 (c) 13 (d) 11
  32. 32.What are the roots of x² − 10x + 21 = 0? (a) 1 and 21 (b) 2 and 8 (c) 1 and 9 (d) 3 and 7
  33. 33.If a + b = 8 and ab = 15, what is a² + b²? (a) 49 (b) 64 (c) 94 (d) 34
  34. 34.If a + b = 18 and ab = 72, what is a² + b²? (a) 324 (b) 252 (c) 468 (d) 180
  35. 35.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6241 (b) 6401 (c) 6571 (d) 6561
  36. 36.What are the roots of x² − 11x + 30 = 0? (a) 1 and 30 (b) 1 and 10 (c) 2 and 15 (d) 5 and 6
  37. 37.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 401 (b) 441 (c) 451 (d) 361
  38. 38.What are the roots of x² − 13x + 42 = 0? (a) 1 and 42 (b) 6 and 7 (c) 2 and 21 (d) 1 and 12
  39. 39.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9984 (b) 10816 (c) 9226 (d) 9216
  40. 40.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1859 (b) 1609 (c) 1369 (d) 1849

Answer key

  1. (x + 5)(x + 6)
  2. 8281
  3. 56
  4. 5929
  5. 1225
  6. 13
  7. 42
  8. 27
  9. 85
  10. 1089
  11. 10
  12. 9
  13. 4225
  14. 81
  15. 4 and 5
  16. 676
  17. (x + 8)(x + 9)
  18. 9025
  19. 16
  20. 16
  21. 17
  22. 7
  23. 100
  24. (x + 4)(x + 8)
  25. 181
  26. 8 and 9
  27. 3364
  28. 10
  29. (x + 6)(x + 7)
  30. (x + 4)(x + 5)
  31. 12
  32. 3 and 7
  33. 34
  34. 180
  35. 6561
  36. 5 and 6
  37. 441
  38. 6 and 7
  39. 9216
  40. 1849

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