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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 173 · 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 + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 1)(x + 4) (c) (x + 1)(x + 6) (d) (x + 2)(x + 3)
  2. 2.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3604 (b) 3844 (c) 3364 (d) 3854
  3. 3.What is the product of the roots of x² − 11x + 30 = 0? (a) 30 (b) 31 (c) 11 (d) 29
  4. 4.If a + b = 14 and ab = 33, what is a² + b²? (a) 196 (b) 130 (c) 262 (d) 163
  5. 5.Which is the correct factorisation of x² + 7x + 10? (a) (x + 2)(x + 5) (b) (x + 1)(x + 10) (c) (x − 2)(x − 5) (d) (x + 1)(x + 6)
  6. 6.What are the roots of x² − 11x + 24 = 0? (a) 3 and 8 (b) 2 and 12 (c) 1 and 24 (d) 1 and 10
  7. 7.If a + b = 17 and ab = 72, what is (a − b)²? (a) 289 (b) 1 (c) 217 (d) 145
  8. 8.What are the roots of x² − 16x + 63 = 0? (a) 3 and 21 (b) 1 and 15 (c) 1 and 63 (d) 7 and 9
  9. 9.If a + b = 16 and ab = 55, what is a² + b²? (a) 256 (b) 366 (c) 201 (d) 146
  10. 10.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1601 (b) 1521 (c) 1681 (d) 1691
  11. 11.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 3)(x + 9) (c) (x + 1)(x + 11) (d) (x + 1)(x + 27)
  12. 12.If a + b = 16 and ab = 55, what is (a − b)²? (a) 256 (b) 36 (c) 146 (d) 6
  13. 13.If a + b = 20 and ab = 96, what is (a − b)²? (a) 16 (b) 4 (c) 400 (d) 208
  14. 14.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x − 2)(x − 4) (c) (x + 2)(x + 4) (d) (x + 1)(x + 8)
  15. 15.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x + 1)(x + 11) (c) (x + 4)(x + 8) (d) (x − 4)(x − 8)
  16. 16.What are the roots of x² − 13x + 42 = 0? (a) 1 and 42 (b) 2 and 21 (c) 1 and 12 (d) 6 and 7
  17. 17.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 72 (c) 17 (d) 18
  18. 18.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 4)(x + 5) (c) (x + 1)(x + 20) (d) (x + 1)(x + 8)
  19. 19.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 4225 (b) 3035 (c) 3025 (d) 3575
  20. 20.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2916 (b) 2126 (c) 2116 (d) 2484
  21. 21.If a + b = 18 and ab = 80, what is a² + b²? (a) 324 (b) 484 (c) 244 (d) 164
  22. 22.If a + b = 17 and ab = 70, what is (a − b)²? (a) 9 (b) 3 (c) 149 (d) 289
  23. 23.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 73 (c) 72 (d) 17
  24. 24.If a + b = 16 and ab = 63, what is (a − b)²? (a) 4 (b) 130 (c) 256 (d) 2
  25. 25.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 202 (c) 301 (d) 400
  26. 26.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 55 (c) 15 (d) 56
  27. 27.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3136 (b) 4106 (c) 4096 (d) 3616
  28. 28.What are the roots of x² − 9x + 14 = 0? (a) 2 and 7 (b) 1 and 8 (c) 3 and 6 (d) 1 and 14
  29. 29.If a + b = 12 and ab = 35, what is (a − b)²? (a) 74 (b) 2 (c) 144 (d) 4
  30. 30.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 841 (b) 971 (c) 901 (d) 961
  31. 31.If a + b = 7 and ab = 12, what is a² + b²? (a) 49 (b) 73 (c) 25 (d) 37
  32. 32.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6734 (b) 6084 (c) 6404 (d) 6724
  33. 33.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 5)(x + 9) (c) (x + 1)(x + 13) (d) (x + 1)(x + 45)
  34. 34.What is the sum of the roots of x² − 12x + 35 = 0? (a) 35 (b) 11 (c) 13 (d) 12
  35. 35.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 1)(x + 12) (c) (x + 4)(x + 9) (d) (x + 1)(x + 36)
  36. 36.If a + b = 10 and ab = 16, what is a² + b²? (a) 100 (b) 132 (c) 84 (d) 68
  37. 37.What is the product of the roots of x² − 7x + 10 = 0? (a) 7 (b) 11 (c) 10 (d) 9
  38. 38.What are the roots of x² − 11x + 30 = 0? (a) 5 and 6 (b) 1 and 10 (c) 1 and 30 (d) 2 and 15
  39. 39.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1936 (b) 1296 (c) 1584 (d) 1306
  40. 40.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 739 (b) 729 (c) 891 (d) 1089

Answer key

  1. (x + 2)(x + 3)
  2. 3844
  3. 30
  4. 130
  5. (x + 2)(x + 5)
  6. 3 and 8
  7. 1
  8. 7 and 9
  9. 146
  10. 1681
  11. (x + 3)(x + 9)
  12. 36
  13. 16
  14. (x + 2)(x + 4)
  15. (x + 4)(x + 8)
  16. 6 and 7
  17. 17
  18. (x + 4)(x + 5)
  19. 3025
  20. 2116
  21. 164
  22. 9
  23. 72
  24. 4
  25. 202
  26. 56
  27. 4096
  28. 2 and 7
  29. 4
  30. 961
  31. 25
  32. 6724
  33. (x + 5)(x + 9)
  34. 12
  35. (x + 4)(x + 9)
  36. 68
  37. 10
  38. 5 and 6
  39. 1296
  40. 729

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