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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 531 · 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.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5051 (b) 4761 (c) 4901 (d) 5041
  2. 2.If a + b = 14 and ab = 40, what is (a − b)²? (a) 116 (b) 196 (c) 6 (d) 36
  3. 3.If a + b = 10 and ab = 16, what is a² + b²? (a) 100 (b) 132 (c) 84 (d) 68
  4. 4.What is the sum of the roots of x² − 9x + 20 = 0? (a) 8 (b) 20 (c) 10 (d) 9
  5. 5.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2126 (b) 2916 (c) 2116 (d) 2484
  6. 6.If a + b = 17 and ab = 70, what is (a − b)²? (a) 149 (b) 289 (c) 9 (d) 3
  7. 7.What are the roots of x² − 9x + 18 = 0? (a) 3 and 6 (b) 1 and 8 (c) 1 and 18 (d) 2 and 9
  8. 8.If a + b = 20 and ab = 96, what is a² + b²? (a) 592 (b) 208 (c) 400 (d) 304
  9. 9.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x − 6)(x − 9) (c) (x + 6)(x + 9) (d) (x + 1)(x + 14)
  10. 10.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 657 (c) 333 (d) 225
  11. 11.If a + b = 13 and ab = 22, what is (a − b)²? (a) 169 (b) 125 (c) 81 (d) 9
  12. 12.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 1225 (b) 875 (c) 635 (d) 625
  13. 13.If a + b = 11 and ab = 24, what is (a − b)²? (a) 73 (b) 25 (c) 5 (d) 121
  14. 14.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 18 (d) 16
  15. 15.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
  16. 16.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3979 (b) 3249 (c) 3969 (d) 3609
  17. 17.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 14) (c) (x + 1)(x + 56) (d) (x + 7)(x + 8)
  18. 18.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 56 (c) 16 (d) 14
  19. 19.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 1)(x + 30) (c) (x + 1)(x + 10) (d) (x + 5)(x + 6)
  20. 20.What is the sum of the roots of x² − 11x + 28 = 0? (a) 10 (b) 11 (c) 12 (d) 28
  21. 21.If a + b = 11 and ab = 30, what is a² + b²? (a) 61 (b) 121 (c) 181 (d) 91
  22. 22.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 451 (b) 401 (c) 361 (d) 441
  23. 23.What is the product of the roots of x² − 12x + 32 = 0? (a) 12 (b) 31 (c) 33 (d) 32
  24. 24.What is the sum of the roots of x² − 6x + 8 = 0? (a) 7 (b) 8 (c) 5 (d) 6
  25. 25.If a + b = 11 and ab = 10, what is a² + b²? (a) 111 (b) 101 (c) 141 (d) 121
  26. 26.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4761 (b) 4899 (c) 4771 (d) 5041
  27. 27.What are the roots of x² − 14x + 48 = 0? (a) 2 and 24 (b) 1 and 13 (c) 1 and 48 (d) 6 and 8
  28. 28.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x − 4)(x − 9) (c) (x + 4)(x + 9) (d) (x + 1)(x + 12)
  29. 29.What are the roots of x² − 8x + 12 = 0? (a) 3 and 4 (b) 2 and 6 (c) 1 and 12 (d) 1 and 7
  30. 30.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 784 (c) 896 (d) 1024
  31. 31.If a + b = 16 and ab = 48, what is a² + b²? (a) 256 (b) 208 (c) 352 (d) 160
  32. 32.What is the sum of the roots of x² − 12x + 32 = 0? (a) 32 (b) 12 (c) 11 (d) 13
  33. 33.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 145 (c) 289 (d) 217
  34. 34.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1774 (b) 1764 (c) 1604 (d) 1444
  35. 35.What are the roots of x² − 8x + 15 = 0? (a) 2 and 6 (b) 1 and 7 (c) 3 and 5 (d) 1 and 15
  36. 36.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1859 (b) 1849 (c) 1369 (d) 1609
  37. 37.What is the sum of the roots of x² − 11x + 30 = 0? (a) 12 (b) 30 (c) 11 (d) 10
  38. 38.If a + b = 21 and ab = 108, what is (a − b)²? (a) 9 (b) 3 (c) 441 (d) 225
  39. 39.If a + b = 10 and ab = 16, what is (a − b)²? (a) 100 (b) 6 (c) 36 (d) 68
  40. 40.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 4096 (b) 3136 (c) 3146 (d) 3584

Answer key

  1. 5041
  2. 36
  3. 68
  4. 9
  5. 2116
  6. 9
  7. 3 and 6
  8. 208
  9. (x + 6)(x + 9)
  10. 225
  11. 81
  12. 625
  13. 25
  14. 17
  15. (x + 8)(x + 9)
  16. 3969
  17. (x + 7)(x + 8)
  18. 15
  19. (x + 5)(x + 6)
  20. 11
  21. 61
  22. 441
  23. 32
  24. 6
  25. 101
  26. 4761
  27. 6 and 8
  28. (x + 4)(x + 9)
  29. 2 and 6
  30. 784
  31. 160
  32. 12
  33. 145
  34. 1764
  35. 3 and 5
  36. 1849
  37. 11
  38. 9
  39. 36
  40. 3136

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