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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 641 · 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 104²? (a) 10016 (b) 9216 (c) 10816 (d) 10826
  2. 2.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x − 5)(x − 8) (c) (x + 1)(x + 12) (d) (x + 5)(x + 8)
  3. 3.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)
  4. 4.If a + b = 11 and ab = 24, what is a² + b²? (a) 169 (b) 73 (c) 121 (d) 97
  5. 5.What is the product of the roots of x² − 9x + 14 = 0? (a) 9 (b) 14 (c) 13 (d) 15
  6. 6.If a + b = 17 and ab = 72, what is a² + b²? (a) 289 (b) 145 (c) 217 (d) 433
  7. 7.If a + b = 15 and ab = 56, what is a² + b²? (a) 337 (b) 113 (c) 225 (d) 169
  8. 8.If a + b = 20 and ab = 96, what is a² + b²? (a) 400 (b) 592 (c) 208 (d) 304
  9. 9.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x + 1)(x + 15) (c) (x − 3)(x − 5) (d) (x + 3)(x + 5)
  10. 10.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 5184 (b) 4624 (c) 4634 (d) 4896
  11. 11.What is the product of the roots of x² − 9x + 20 = 0? (a) 21 (b) 20 (c) 19 (d) 9
  12. 12.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9975 (b) 9025 (c) 11025 (d) 9035
  13. 13.What is the sum of the roots of x² − 7x + 12 = 0? (a) 12 (b) 7 (c) 6 (d) 8
  14. 14.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 1)(x + 56) (c) (x + 7)(x + 8) (d) (x − 7)(x − 8)
  15. 15.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6391 (b) 6889 (c) 5939 (d) 5929
  16. 16.What is the sum of the roots of x² − 13x + 40 = 0? (a) 40 (b) 12 (c) 13 (d) 14
  17. 17.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8291 (b) 8101 (c) 8281 (d) 7921
  18. 18.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 14 (c) 15 (d) 16
  19. 19.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 1)(x + 13) (c) (x + 5)(x + 9) (d) (x + 1)(x + 45)
  20. 20.If a + b = 11 and ab = 28, what is (a − b)²? (a) 9 (b) 121 (c) 3 (d) 65
  21. 21.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 2025 (b) 1225 (c) 1235 (d) 1575
  22. 22.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2401 (b) 2611 (c) 2601 (d) 2501
  23. 23.If a + b = 14 and ab = 33, what is a² + b²? (a) 130 (b) 196 (c) 163 (d) 262
  24. 24.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 971 (b) 901 (c) 841 (d) 961
  25. 25.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 73 (c) 71 (d) 72
  26. 26.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 7744 (b) 8104 (c) 8464 (d) 8474
  27. 27.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 891 (b) 739 (c) 729 (d) 1089
  28. 28.What are the roots of x² − 9x + 20 = 0? (a) 2 and 10 (b) 1 and 20 (c) 1 and 8 (d) 4 and 5
  29. 29.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 266 (b) 384 (c) 256 (d) 576
  30. 30.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 16 (c) 17 (d) 18
  31. 31.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4899 (b) 5041 (c) 4761 (d) 4771
  32. 32.If a + b = 15 and ab = 44, what is (a − b)²? (a) 137 (b) 7 (c) 49 (d) 225
  33. 33.What is the sum of the roots of x² − 12x + 32 = 0? (a) 13 (b) 11 (c) 32 (d) 12
  34. 34.What are the roots of x² − 8x + 15 = 0? (a) 2 and 6 (b) 3 and 5 (c) 1 and 7 (d) 1 and 15
  35. 35.What is the sum of the roots of x² − 6x + 8 = 0? (a) 5 (b) 8 (c) 6 (d) 7
  36. 36.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x − 4)(x − 8) (c) (x + 1)(x + 11) (d) (x + 4)(x + 8)
  37. 37.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)
  38. 38.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 909 (b) 1089 (c) 1099 (d) 729
  39. 39.What is the sum of the roots of x² − 12x + 35 = 0? (a) 35 (b) 13 (c) 12 (d) 11
  40. 40.If a + b = 16 and ab = 48, what is a² + b²? (a) 208 (b) 160 (c) 352 (d) 256

Answer key

  1. 10816
  2. (x + 5)(x + 8)
  3. (x + 8)(x + 9)
  4. 73
  5. 14
  6. 145
  7. 113
  8. 208
  9. (x + 3)(x + 5)
  10. 4624
  11. 20
  12. 9025
  13. 7
  14. (x + 7)(x + 8)
  15. 5929
  16. 13
  17. 8281
  18. 15
  19. (x + 5)(x + 9)
  20. 9
  21. 1225
  22. 2601
  23. 130
  24. 961
  25. 72
  26. 8464
  27. 729
  28. 4 and 5
  29. 256
  30. 17
  31. 4761
  32. 49
  33. 12
  34. 3 and 5
  35. 6
  36. (x + 4)(x + 8)
  37. (x + 2)(x + 4)
  38. 1089
  39. 12
  40. 160

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