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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 752 · 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 68²? (a) 4634 (b) 4896 (c) 4624 (d) 5184
  2. 2.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 5776 (b) 6384 (c) 7056 (d) 5786
  3. 3.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 8836 (b) 7406 (c) 7396 (d) 8084
  4. 4.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 7225 (b) 9025 (c) 8125 (d) 9035
  5. 5.If a + b = 15 and ab = 36, what is a² + b²? (a) 153 (b) 189 (c) 297 (d) 225
  6. 6.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x + 3)(x + 4) (c) (x + 1)(x + 6) (d) (x − 3)(x − 4)
  7. 7.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5041 (b) 5051 (c) 4761 (d) 4901
  8. 8.If a + b = 18 and ab = 77, what is (a − b)²? (a) 16 (b) 324 (c) 170 (d) 4
  9. 9.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6396 (b) 6084 (c) 6094 (d) 6724
  10. 10.What is the sum of the roots of x² − 11x + 28 = 0? (a) 12 (b) 11 (c) 10 (d) 28
  11. 11.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 5)(x + 6) (c) (x + 1)(x + 10) (d) (x − 5)(x − 6)
  12. 12.What is the product of the roots of x² − 13x + 42 = 0? (a) 13 (b) 41 (c) 42 (d) 43
  13. 13.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 361 (b) 401 (c) 441 (d) 451
  14. 14.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x + 1)(x + 13) (c) (x − 5)(x − 9) (d) (x + 5)(x + 9)
  15. 15.If a + b = 11 and ab = 10, what is (a − b)²? (a) 121 (b) 9 (c) 81 (d) 101
  16. 16.If a + b = 19 and ab = 84, what is (a − b)²? (a) 361 (b) 5 (c) 25 (d) 193
  17. 17.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 1)(x + 27) (c) (x + 1)(x + 11) (d) (x + 3)(x + 9)
  18. 18.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 13 (c) 14 (d) 15
  19. 19.What are the roots of x² − 7x + 12 = 0? (a) 2 and 6 (b) 1 and 6 (c) 3 and 4 (d) 1 and 12
  20. 20.If a + b = 15 and ab = 54, what is (a − b)²? (a) 117 (b) 9 (c) 3 (d) 225
  21. 21.What are the roots of x² − 16x + 63 = 0? (a) 1 and 15 (b) 1 and 63 (c) 7 and 9 (d) 3 and 21
  22. 22.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1859 (b) 1849 (c) 1369 (d) 1609
  23. 23.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x − 7)(x − 8) (d) (x + 7)(x + 8)
  24. 24.What are the roots of x² − 11x + 30 = 0? (a) 5 and 6 (b) 1 and 30 (c) 2 and 15 (d) 1 and 10
  25. 25.What are the roots of x² − 10x + 21 = 0? (a) 1 and 21 (b) 3 and 7 (c) 1 and 9 (d) 2 and 8
  26. 26.Which is the correct factorisation of x² + 13x + 36? (a) (x + 4)(x + 9) (b) (x − 4)(x − 9) (c) (x + 1)(x + 12) (d) (x + 1)(x + 36)
  27. 27.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 3)(x + 6) (c) (x + 1)(x + 8) (d) (x + 1)(x + 18)
  28. 28.If a + b = 10 and ab = 16, what is (a − b)²? (a) 68 (b) 100 (c) 36 (d) 6
  29. 29.If a + b = 14 and ab = 24, what is (a − b)²? (a) 148 (b) 100 (c) 196 (d) 10
  30. 30.If a + b = 9 and ab = 18, what is a² + b²? (a) 117 (b) 81 (c) 63 (d) 45
  31. 31.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 1 (c) 181 (d) 361
  32. 32.If a + b = 5 and ab = 4, what is (a − b)²? (a) 17 (b) 9 (c) 3 (d) 25
  33. 33.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 409 (b) 529 (c) 539 (d) 289
  34. 34.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8116 (b) 7396 (c) 8836 (d) 8846
  35. 35.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 14 (c) 15 (d) 16
  36. 36.If a + b = 15 and ab = 50, what is (a − b)²? (a) 125 (b) 225 (c) 25 (d) 5
  37. 37.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3844 (b) 3364 (c) 3374 (d) 3596
  38. 38.If a + b = 7 and ab = 12, what is (a − b)²? (a) 1 (b) 49 (c) 25 (d) 37
  39. 39.If a + b = 19 and ab = 88, what is a² + b²? (a) 185 (b) 273 (c) 361 (d) 537
  40. 40.If a + b = 12 and ab = 20, what is (a − b)²? (a) 64 (b) 104 (c) 8 (d) 144

Answer key

  1. 4624
  2. 5776
  3. 7396
  4. 9025
  5. 153
  6. (x + 3)(x + 4)
  7. 5041
  8. 16
  9. 6084
  10. 11
  11. (x + 5)(x + 6)
  12. 42
  13. 441
  14. (x + 5)(x + 9)
  15. 81
  16. 25
  17. (x + 3)(x + 9)
  18. 14
  19. 3 and 4
  20. 9
  21. 7 and 9
  22. 1849
  23. (x + 7)(x + 8)
  24. 5 and 6
  25. 3 and 7
  26. (x + 4)(x + 9)
  27. (x + 3)(x + 6)
  28. 36
  29. 100
  30. 45
  31. 1
  32. 9
  33. 529
  34. 8836
  35. 15
  36. 25
  37. 3364
  38. 1
  39. 185
  40. 64

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