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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 238 · 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 = 11 and ab = 18, what is a² + b²? (a) 157 (b) 103 (c) 121 (d) 85
  2. 2.If a + b = 20 and ab = 96, what is a² + b²? (a) 304 (b) 208 (c) 400 (d) 592
  3. 3.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x + 1)(x + 54) (c) (x − 6)(x − 9) (d) (x + 6)(x + 9)
  4. 4.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 1)(x + 48) (c) (x − 6)(x − 8) (d) (x + 6)(x + 8)
  5. 5.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 2 and 28 (c) 7 and 8 (d) 1 and 14
  6. 6.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x + 1)(x + 16) (c) (x − 2)(x − 8) (d) (x + 2)(x + 8)
  7. 7.If a + b = 12 and ab = 11, what is (a − b)²? (a) 144 (b) 122 (c) 10 (d) 100
  8. 8.Which is the correct factorisation of x² + 5x + 6? (a) (x + 2)(x + 3) (b) (x − 2)(x − 3) (c) (x + 1)(x + 4) (d) (x + 1)(x + 6)
  9. 9.What is the sum of the roots of x² − 7x + 12 = 0? (a) 7 (b) 8 (c) 6 (d) 12
  10. 10.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 271 (c) 181 (d) 361
  11. 11.What is the product of the roots of x² − 11x + 30 = 0? (a) 30 (b) 31 (c) 11 (d) 29
  12. 12.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1946 (b) 1296 (c) 1936 (d) 1616
  13. 13.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 5625 (b) 7235 (c) 6425 (d) 7225
  14. 14.What are the roots of x² − 9x + 20 = 0? (a) 4 and 5 (b) 2 and 10 (c) 1 and 20 (d) 1 and 8
  15. 15.What are the roots of x² − 16x + 63 = 0? (a) 7 and 9 (b) 3 and 21 (c) 1 and 15 (d) 1 and 63
  16. 16.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3854 (b) 3844 (c) 3604 (d) 3364
  17. 17.If a + b = 8 and ab = 15, what is (a − b)²? (a) 4 (b) 64 (c) 2 (d) 34
  18. 18.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2525 (b) 2025 (c) 3035 (d) 3025
  19. 19.If a + b = 6 and ab = 8, what is a² + b²? (a) 52 (b) 36 (c) 28 (d) 20
  20. 20.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8096 (b) 8464 (c) 7754 (d) 7744
  21. 21.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 625 (b) 1225 (c) 1235 (d) 925
  22. 22.What is the sum of the roots of x² − 13x + 36 = 0? (a) 14 (b) 12 (c) 13 (d) 36
  23. 23.What is the product of the roots of x² − 14x + 45 = 0? (a) 14 (b) 45 (c) 44 (d) 46
  24. 24.If a + b = 10 and ab = 21, what is (a − b)²? (a) 16 (b) 100 (c) 4 (d) 58
  25. 25.What is the sum of the roots of x² − 13x + 40 = 0? (a) 40 (b) 13 (c) 14 (d) 12
  26. 26.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 14 (c) 16 (d) 56
  27. 27.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x − 5)(x − 8) (c) (x + 5)(x + 8) (d) (x + 1)(x + 40)
  28. 28.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3731 (b) 3601 (c) 3721 (d) 3481
  29. 29.If a + b = 18 and ab = 77, what is (a − b)²? (a) 16 (b) 324 (c) 170 (d) 4
  30. 30.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6391 (b) 5929 (c) 6889 (d) 5939
  31. 31.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4225 (b) 4875 (c) 5625 (d) 4235
  32. 32.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4366 (b) 4356 (c) 5476 (d) 4884
  33. 33.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x − 5)(x − 6) (c) (x + 1)(x + 10) (d) (x + 1)(x + 30)
  34. 34.If a + b = 11 and ab = 24, what is a² + b²? (a) 169 (b) 73 (c) 121 (d) 97
  35. 35.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 686 (b) 884 (c) 676 (d) 1156
  36. 36.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6396 (b) 6724 (c) 6094 (d) 6084
  37. 37.If a + b = 10 and ab = 16, what is a² + b²? (a) 84 (b) 100 (c) 132 (d) 68
  38. 38.If a + b = 18 and ab = 80, what is (a − b)²? (a) 2 (b) 324 (c) 4 (d) 164
  39. 39.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 6)(x + 7) (c) (x − 6)(x − 7) (d) (x + 1)(x + 42)
  40. 40.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 1024 (b) 794 (c) 896 (d) 784

Answer key

  1. 85
  2. 208
  3. (x + 6)(x + 9)
  4. (x + 6)(x + 8)
  5. 7 and 8
  6. (x + 2)(x + 8)
  7. 100
  8. (x + 2)(x + 3)
  9. 7
  10. 181
  11. 30
  12. 1936
  13. 7225
  14. 4 and 5
  15. 7 and 9
  16. 3844
  17. 4
  18. 3025
  19. 20
  20. 7744
  21. 1225
  22. 13
  23. 45
  24. 16
  25. 13
  26. 15
  27. (x + 5)(x + 8)
  28. 3721
  29. 16
  30. 5929
  31. 4225
  32. 4356
  33. (x + 5)(x + 6)
  34. 73
  35. 676
  36. 6084
  37. 68
  38. 4
  39. (x + 6)(x + 7)
  40. 784

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