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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 270 · 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 55²? (a) 2525 (b) 3035 (c) 2025 (d) 3025
  2. 2.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 6) (c) (x + 3)(x + 4) (d) (x + 1)(x + 12)
  3. 3.What is the sum of the roots of x² − 9x + 18 = 0? (a) 9 (b) 8 (c) 10 (d) 18
  4. 4.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1616 (b) 1296 (c) 1946 (d) 1936
  5. 5.If a + b = 19 and ab = 84, what is a² + b²? (a) 361 (b) 193 (c) 277 (d) 529
  6. 6.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 14) (c) (x + 6)(x + 9) (d) (x + 1)(x + 54)
  7. 7.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1521 (b) 1601 (c) 1681 (d) 1691
  8. 8.What are the roots of x² − 10x + 16 = 0? (a) 2 and 8 (b) 3 and 7 (c) 1 and 9 (d) 1 and 16
  9. 9.If a + b = 14 and ab = 45, what is (a − b)²? (a) 16 (b) 4 (c) 106 (d) 196
  10. 10.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 784 (b) 904 (c) 1034 (d) 1024
  11. 11.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2304 (b) 2714 (c) 2504 (d) 2704
  12. 12.If a + b = 18 and ab = 80, what is (a − b)²? (a) 164 (b) 2 (c) 324 (d) 4
  13. 13.What is the sum of the roots of x² − 9x + 14 = 0? (a) 8 (b) 14 (c) 9 (d) 10
  14. 14.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 909 (b) 729 (c) 1099 (d) 1089
  15. 15.What is the sum of the roots of x² − 6x + 8 = 0? (a) 6 (b) 7 (c) 8 (d) 5
  16. 16.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x − 7)(x − 9) (c) (x + 7)(x + 9) (d) (x + 1)(x + 63)
  17. 17.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 5625 (b) 7225 (c) 6425 (d) 7235
  18. 18.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 15 (c) 16 (d) 14
  19. 19.If a + b = 16 and ab = 63, what is a² + b²? (a) 382 (b) 256 (c) 130 (d) 193
  20. 20.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1444 (b) 1596 (c) 1764 (d) 1454
  21. 21.What are the roots of x² − 5x + 6 = 0? (a) 1 and 6 (b) 2 and 4 (c) 2 and 3 (d) 1 and 4
  22. 22.If a + b = 11 and ab = 18, what is a² + b²? (a) 85 (b) 103 (c) 157 (d) 121
  23. 23.Which is the correct factorisation of x² + 5x + 6? (a) (x + 2)(x + 3) (b) (x + 1)(x + 4) (c) (x − 2)(x − 3) (d) (x + 1)(x + 6)
  24. 24.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4761 (b) 4899 (c) 5041 (d) 4771
  25. 25.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3721 (b) 3481 (c) 3731 (d) 3601
  26. 26.What is the product of the roots of x² − 13x + 42 = 0? (a) 43 (b) 41 (c) 13 (d) 42
  27. 27.If a + b = 13 and ab = 40, what is a² + b²? (a) 169 (b) 249 (c) 129 (d) 89
  28. 28.If a + b = 21 and ab = 108, what is a² + b²? (a) 657 (b) 333 (c) 225 (d) 441
  29. 29.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 1)(x + 8) (c) (x + 1)(x + 14) (d) (x + 2)(x + 7)
  30. 30.What are the roots of x² − 13x + 36 = 0? (a) 2 and 18 (b) 1 and 12 (c) 4 and 9 (d) 1 and 36
  31. 31.What is the sum of the roots of x² − 11x + 24 = 0? (a) 11 (b) 12 (c) 24 (d) 10
  32. 32.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3979 (b) 3609 (c) 3969 (d) 3249
  33. 33.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 9025 (b) 11025 (c) 10025 (d) 11035
  34. 34.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x − 8)(x − 9)
  35. 35.If a + b = 9 and ab = 8, what is a² + b²? (a) 81 (b) 73 (c) 97 (d) 65
  36. 36.If a + b = 20 and ab = 96, what is a² + b²? (a) 592 (b) 208 (c) 304 (d) 400
  37. 37.If a + b = 13 and ab = 42, what is a² + b²? (a) 169 (b) 253 (c) 127 (d) 85
  38. 38.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 18 (c) 16 (d) 72
  39. 39.If a + b = 5 and ab = 4, what is a² + b²? (a) 17 (b) 21 (c) 25 (d) 33
  40. 40.If a + b = 13 and ab = 12, what is a² + b²? (a) 157 (b) 193 (c) 169 (d) 145

Answer key

  1. 3025
  2. (x + 3)(x + 4)
  3. 9
  4. 1936
  5. 193
  6. (x + 6)(x + 9)
  7. 1681
  8. 2 and 8
  9. 16
  10. 1024
  11. 2704
  12. 4
  13. 9
  14. 1089
  15. 6
  16. (x + 7)(x + 9)
  17. 7225
  18. 15
  19. 130
  20. 1444
  21. 2 and 3
  22. 85
  23. (x + 2)(x + 3)
  24. 4761
  25. 3721
  26. 42
  27. 89
  28. 225
  29. (x + 2)(x + 7)
  30. 4 and 9
  31. 11
  32. 3969
  33. 11025
  34. (x + 8)(x + 9)
  35. 65
  36. 208
  37. 85
  38. 17
  39. 17
  40. 145

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