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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 184 · 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 27²? (a) 729 (b) 891 (c) 739 (d) 1089
  2. 2.What are the roots of x² − 14x + 48 = 0? (a) 6 and 8 (b) 2 and 24 (c) 1 and 48 (d) 1 and 13
  3. 3.What is the product of the roots of x² − 16x + 63 = 0? (a) 16 (b) 62 (c) 64 (d) 63
  4. 4.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6396 (b) 6724 (c) 6084 (d) 6094
  5. 5.If a + b = 16 and ab = 55, what is a² + b²? (a) 256 (b) 201 (c) 366 (d) 146
  6. 6.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 361 (b) 451 (c) 401 (d) 441
  7. 7.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 1)(x + 5) (c) (x − 2)(x − 4) (d) (x + 2)(x + 4)
  8. 8.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1764 (b) 1444 (c) 1774 (d) 1604
  9. 9.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 1)(x + 72) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
  10. 10.What are the roots of x² − 12x + 32 = 0? (a) 1 and 32 (b) 1 and 11 (c) 2 and 16 (d) 4 and 8
  11. 11.If a + b = 18 and ab = 80, what is a² + b²? (a) 484 (b) 244 (c) 324 (d) 164
  12. 12.If a + b = 20 and ab = 99, what is (a − b)²? (a) 202 (b) 2 (c) 4 (d) 400
  13. 13.Which is the correct factorisation of x² + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 1)(x + 9) (c) (x + 1)(x + 16) (d) (x + 2)(x + 8)
  14. 14.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 16 (c) 15 (d) 14
  15. 15.If a + b = 7 and ab = 10, what is a² + b²? (a) 29 (b) 49 (c) 69 (d) 39
  16. 16.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 851 (b) 841 (c) 961 (d) 899
  17. 17.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 18 (c) 72 (d) 17
  18. 18.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 901 (b) 841 (c) 961 (d) 971
  19. 19.If a + b = 17 and ab = 66, what is a² + b²? (a) 289 (b) 421 (c) 157 (d) 223
  20. 20.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)
  21. 21.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6571 (b) 6241 (c) 6401 (d) 6561
  22. 22.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1296 (b) 1946 (c) 1616 (d) 1936
  23. 23.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 5194 (b) 4624 (c) 5184 (d) 4904
  24. 24.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8091 (b) 7579 (c) 7569 (d) 8649
  25. 25.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 11025 (b) 9025 (c) 9035 (d) 9975
  26. 26.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 416 (b) 576 (c) 586 (d) 256
  27. 27.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6241 (b) 6561 (c) 6399 (d) 6251
  28. 28.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6899 (b) 6409 (c) 6889 (d) 5929
  29. 29.If a + b = 18 and ab = 72, what is (a − b)²? (a) 180 (b) 6 (c) 36 (d) 324
  30. 30.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4761 (b) 4901 (c) 5041 (d) 5051
  31. 31.If a + b = 15 and ab = 56, what is (a − b)²? (a) 113 (b) 225 (c) 169 (d) 1
  32. 32.If a + b = 14 and ab = 48, what is a² + b²? (a) 100 (b) 292 (c) 148 (d) 196
  33. 33.If a + b = 19 and ab = 84, what is a² + b²? (a) 193 (b) 361 (c) 529 (d) 277
  34. 34.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1099 (b) 729 (c) 1089 (d) 909
  35. 35.Which is the correct factorisation of x² + 12x + 35? (a) (x + 5)(x + 7) (b) (x − 5)(x − 7) (c) (x + 1)(x + 35) (d) (x + 1)(x + 11)
  36. 36.What are the roots of x² − 10x + 16 = 0? (a) 1 and 16 (b) 1 and 9 (c) 2 and 8 (d) 3 and 7
  37. 37.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 181 (c) 541 (d) 361
  38. 38.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)
  39. 39.If a + b = 15 and ab = 36, what is (a − b)²? (a) 153 (b) 81 (c) 9 (d) 225
  40. 40.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x − 3)(x − 6) (c) (x + 1)(x + 18) (d) (x + 3)(x + 6)

Answer key

  1. 729
  2. 6 and 8
  3. 63
  4. 6084
  5. 146
  6. 441
  7. (x + 2)(x + 4)
  8. 1764
  9. (x + 8)(x + 9)
  10. 4 and 8
  11. 164
  12. 4
  13. (x + 2)(x + 8)
  14. 15
  15. 29
  16. 841
  17. 17
  18. 961
  19. 157
  20. (x + 7)(x + 9)
  21. 6561
  22. 1936
  23. 5184
  24. 7569
  25. 9025
  26. 576
  27. 6241
  28. 6889
  29. 36
  30. 5041
  31. 1
  32. 100
  33. 193
  34. 1089
  35. (x + 5)(x + 7)
  36. 2 and 8
  37. 181
  38. (x + 4)(x + 8)
  39. 81
  40. (x + 3)(x + 6)

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