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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 138 · 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 = 18 and ab = 80, what is a² + b²? (a) 164 (b) 244 (c) 324 (d) 484
  2. 2.What are the roots of x² − 16x + 63 = 0? (a) 1 and 63 (b) 1 and 15 (c) 7 and 9 (d) 3 and 21
  3. 3.If a + b = 19 and ab = 84, what is (a − b)²? (a) 193 (b) 25 (c) 361 (d) 5
  4. 4.What are the roots of x² − 8x + 12 = 0? (a) 3 and 4 (b) 1 and 7 (c) 1 and 12 (d) 2 and 6
  5. 5.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 56) (c) (x + 1)(x + 14) (d) (x − 7)(x − 8)
  6. 6.Which is the correct factorisation of x² + 12x + 27? (a) (x + 3)(x + 9) (b) (x − 3)(x − 9) (c) (x + 1)(x + 11) (d) (x + 1)(x + 27)
  7. 7.If a + b = 17 and ab = 70, what is a² + b²? (a) 219 (b) 429 (c) 149 (d) 289
  8. 8.What is the sum of the roots of x² − 11x + 28 = 0? (a) 12 (b) 11 (c) 10 (d) 28
  9. 9.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 361 (c) 541 (d) 181
  10. 10.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 676 (b) 916 (c) 1156 (d) 1166
  11. 11.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 529 (b) 289 (c) 409 (d) 539
  12. 12.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 1 and 16 (c) 2 and 36 (d) 8 and 9
  13. 13.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 15 (c) 56 (d) 14
  14. 14.If a + b = 16 and ab = 63, what is (a − b)²? (a) 4 (b) 130 (c) 256 (d) 2
  15. 15.What is the product of the roots of x² − 10x + 21 = 0? (a) 20 (b) 21 (c) 10 (d) 22
  16. 16.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 181 (c) 361 (d) 1
  17. 17.What are the roots of x² − 8x + 15 = 0? (a) 1 and 15 (b) 1 and 7 (c) 2 and 6 (d) 3 and 5
  18. 18.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3721 (b) 3481 (c) 3491 (d) 3599
  19. 19.If a + b = 5 and ab = 6, what is a² + b²? (a) 25 (b) 13 (c) 37 (d) 19
  20. 20.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 1024 (b) 896 (c) 784 (d) 794
  21. 21.If a + b = 16 and ab = 48, what is a² + b²? (a) 160 (b) 256 (c) 352 (d) 208
  22. 22.What is the sum of the roots of x² − 6x + 8 = 0? (a) 7 (b) 8 (c) 5 (d) 6
  23. 23.If a + b = 20 and ab = 99, what is a² + b²? (a) 301 (b) 400 (c) 598 (d) 202
  24. 24.What is the product of the roots of x² − 15x + 54 = 0? (a) 55 (b) 54 (c) 53 (d) 15
  25. 25.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 8)(x + 9) (c) (x + 1)(x + 16) (d) (x − 8)(x − 9)
  26. 26.What is the sum of the roots of x² − 7x + 12 = 0? (a) 8 (b) 6 (c) 7 (d) 12
  27. 27.If a + b = 9 and ab = 14, what is (a − b)²? (a) 81 (b) 25 (c) 5 (d) 53
  28. 28.If a + b = 13 and ab = 30, what is (a − b)²? (a) 109 (b) 7 (c) 49 (d) 169
  29. 29.What is the product of the roots of x² − 7x + 10 = 0? (a) 11 (b) 10 (c) 9 (d) 7
  30. 30.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2916 (b) 2116 (c) 2516 (d) 2926
  31. 31.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 6375 (b) 5625 (c) 5635 (d) 7225
  32. 32.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 16 (c) 17 (d) 18
  33. 33.What are the roots of x² − 9x + 18 = 0? (a) 3 and 6 (b) 2 and 9 (c) 1 and 18 (d) 1 and 8
  34. 34.If a + b = 7 and ab = 10, what is (a − b)²? (a) 3 (b) 49 (c) 9 (d) 29
  35. 35.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 7406 (b) 7396 (c) 8084 (d) 8836
  36. 36.If a + b = 19 and ab = 84, what is a² + b²? (a) 361 (b) 529 (c) 193 (d) 277
  37. 37.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 1)(x + 9) (c) (x + 1)(x + 24) (d) (x + 4)(x + 6)
  38. 38.What is the sum of the roots of x² − 5x + 6 = 0? (a) 7 (b) 5 (c) 6 (d) 4
  39. 39.If a + b = 15 and ab = 50, what is (a − b)²? (a) 225 (b) 125 (c) 25 (d) 5
  40. 40.If a + b = 19 and ab = 88, what is (a − b)²? (a) 3 (b) 361 (c) 185 (d) 9

Answer key

  1. 164
  2. 7 and 9
  3. 25
  4. 2 and 6
  5. (x + 7)(x + 8)
  6. (x + 3)(x + 9)
  7. 149
  8. 11
  9. 181
  10. 1156
  11. 529
  12. 8 and 9
  13. 15
  14. 4
  15. 21
  16. 1
  17. 3 and 5
  18. 3481
  19. 13
  20. 784
  21. 160
  22. 6
  23. 202
  24. 54
  25. (x + 8)(x + 9)
  26. 7
  27. 25
  28. 49
  29. 10
  30. 2916
  31. 5625
  32. 17
  33. 3 and 6
  34. 9
  35. 7396
  36. 193
  37. (x + 4)(x + 6)
  38. 5
  39. 25
  40. 9

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