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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 846 · 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 = 13 and ab = 36, what is a² + b²? (a) 97 (b) 169 (c) 133 (d) 241
  2. 2.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 4489 (b) 4909 (c) 5339 (d) 5329
  3. 3.What are the roots of x² − 15x + 56 = 0? (a) 2 and 28 (b) 7 and 8 (c) 1 and 14 (d) 1 and 56
  4. 4.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 4 (c) 400 (d) 202
  5. 5.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x − 8)(x − 9) (c) (x + 1)(x + 72) (d) (x + 1)(x + 16)
  6. 6.What is the sum of the roots of x² − 5x + 6 = 0? (a) 6 (b) 4 (c) 7 (d) 5
  7. 7.What are the roots of x² − 5x + 6 = 0? (a) 2 and 3 (b) 1 and 4 (c) 1 and 6 (d) 2 and 4
  8. 8.If a + b = 17 and ab = 70, what is a² + b²? (a) 149 (b) 429 (c) 219 (d) 289
  9. 9.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x − 7)(x − 9) (c) (x + 1)(x + 15) (d) (x + 7)(x + 9)
  10. 10.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 13) (c) (x + 1)(x + 48) (d) (x + 6)(x + 8)
  11. 11.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 18 (d) 16
  12. 12.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6571 (b) 6561 (c) 6241 (d) 6401
  13. 13.If a + b = 21 and ab = 108, what is a² + b²? (a) 657 (b) 441 (c) 225 (d) 333
  14. 14.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 971 (b) 961 (c) 901 (d) 841
  15. 15.What are the roots of x² − 11x + 30 = 0? (a) 2 and 15 (b) 5 and 6 (c) 1 and 10 (d) 1 and 30
  16. 16.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 235 (b) 625 (c) 375 (d) 225
  17. 17.If a + b = 19 and ab = 84, what is a² + b²? (a) 193 (b) 529 (c) 277 (d) 361
  18. 18.What is the product of the roots of x² − 15x + 54 = 0? (a) 15 (b) 55 (c) 53 (d) 54
  19. 19.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6724 (b) 6396 (c) 6094 (d) 6084
  20. 20.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 17 (c) 73 (d) 72
  21. 21.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 9604 (b) 10004 (c) 10404 (d) 10414
  22. 22.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x − 4)(x − 8) (c) (x + 1)(x + 11) (d) (x + 4)(x + 8)
  23. 23.If a + b = 14 and ab = 24, what is (a − b)²? (a) 196 (b) 148 (c) 100 (d) 10
  24. 24.What is the product of the roots of x² − 11x + 18 = 0? (a) 19 (b) 17 (c) 11 (d) 18
  25. 25.What is the sum of the roots of x² − 14x + 48 = 0? (a) 15 (b) 13 (c) 48 (d) 14
  26. 26.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6899 (b) 6889 (c) 5929 (d) 6409
  27. 27.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2504 (b) 2714 (c) 2704 (d) 2304
  28. 28.What is the sum of the roots of x² − 8x + 15 = 0? (a) 9 (b) 15 (c) 7 (d) 8
  29. 29.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 7931 (b) 7921 (c) 8281 (d) 8099
  30. 30.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 401 (b) 361 (c) 451 (d) 441
  31. 31.What are the roots of x² − 15x + 54 = 0? (a) 1 and 54 (b) 2 and 27 (c) 1 and 14 (d) 6 and 9
  32. 32.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3575 (b) 4225 (c) 3035 (d) 3025
  33. 33.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 5184 (b) 4624 (c) 5194 (d) 4904
  34. 34.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 7)(x + 8) (c) (x + 1)(x + 56) (d) (x + 1)(x + 14)
  35. 35.What are the roots of x² − 8x + 15 = 0? (a) 3 and 5 (b) 1 and 15 (c) 2 and 6 (d) 1 and 7
  36. 36.If a + b = 9 and ab = 20, what is (a − b)²? (a) 61 (b) 41 (c) 1 (d) 81
  37. 37.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6251 (b) 6399 (c) 6561 (d) 6241
  38. 38.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3591 (b) 3969 (c) 3259 (d) 3249
  39. 39.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x − 3)(x − 6) (c) (x + 3)(x + 6) (d) (x + 1)(x + 8)
  40. 40.If a + b = 16 and ab = 63, what is a² + b²? (a) 193 (b) 130 (c) 382 (d) 256

Answer key

  1. 97
  2. 5329
  3. 7 and 8
  4. 4
  5. (x + 8)(x + 9)
  6. 5
  7. 2 and 3
  8. 149
  9. (x + 7)(x + 9)
  10. (x + 6)(x + 8)
  11. 17
  12. 6561
  13. 225
  14. 961
  15. 5 and 6
  16. 225
  17. 193
  18. 54
  19. 6084
  20. 72
  21. 10404
  22. (x + 4)(x + 8)
  23. 100
  24. 18
  25. 14
  26. 6889
  27. 2704
  28. 8
  29. 7921
  30. 441
  31. 6 and 9
  32. 3025
  33. 5184
  34. (x + 7)(x + 8)
  35. 3 and 5
  36. 1
  37. 6241
  38. 3249
  39. (x + 3)(x + 6)
  40. 130

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