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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 516 · 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 46²? (a) 2126 (b) 2916 (c) 2116 (d) 2484
  2. 2.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8099 (b) 8281 (c) 7921 (d) 7931
  3. 3.What are the roots of x² − 9x + 20 = 0? (a) 1 and 8 (b) 1 and 20 (c) 2 and 10 (d) 4 and 5
  4. 4.If a + b = 16 and ab = 60, what is a² + b²? (a) 256 (b) 136 (c) 196 (d) 376
  5. 5.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 2 and 28 (c) 1 and 14 (d) 7 and 8
  6. 6.What is the product of the roots of x² − 11x + 18 = 0? (a) 11 (b) 19 (c) 18 (d) 17
  7. 7.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 235 (b) 625 (c) 225 (d) 375
  8. 8.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 361 (c) 181 (d) 1
  9. 9.If a + b = 8 and ab = 15, what is a² + b²? (a) 64 (b) 49 (c) 34 (d) 94
  10. 10.What is the product of the roots of x² − 10x + 16 = 0? (a) 10 (b) 15 (c) 17 (d) 16
  11. 11.If a + b = 17 and ab = 72, what is a² + b²? (a) 217 (b) 145 (c) 433 (d) 289
  12. 12.What is the sum of the roots of x² − 15x + 54 = 0? (a) 16 (b) 14 (c) 15 (d) 54
  13. 13.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 1 and 16 (c) 1 and 72 (d) 8 and 9
  14. 14.If a + b = 19 and ab = 88, what is a² + b²? (a) 273 (b) 537 (c) 361 (d) 185
  15. 15.If a + b = 5 and ab = 4, what is (a − b)²? (a) 3 (b) 17 (c) 25 (d) 9
  16. 16.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4875 (b) 5625 (c) 4235 (d) 4225
  17. 17.What is the sum of the roots of x² − 8x + 12 = 0? (a) 12 (b) 9 (c) 7 (d) 8
  18. 18.If a + b = 15 and ab = 56, what is (a − b)²? (a) 225 (b) 113 (c) 169 (d) 1
  19. 19.If a + b = 19 and ab = 84, what is a² + b²? (a) 529 (b) 361 (c) 193 (d) 277
  20. 20.If a + b = 15 and ab = 44, what is a² + b²? (a) 225 (b) 313 (c) 181 (d) 137
  21. 21.What is the sum of the roots of x² − 5x + 6 = 0? (a) 5 (b) 7 (c) 4 (d) 6
  22. 22.If a + b = 17 and ab = 70, what is a² + b²? (a) 219 (b) 289 (c) 429 (d) 149
  23. 23.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x − 4)(x − 9) (c) (x + 1)(x + 12) (d) (x + 4)(x + 9)
  24. 24.If a + b = 18 and ab = 72, what is (a − b)²? (a) 324 (b) 36 (c) 6 (d) 180
  25. 25.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 541 (c) 271 (d) 181
  26. 26.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 851 (b) 899 (c) 841 (d) 961
  27. 27.What are the roots of x² − 5x + 6 = 0? (a) 1 and 4 (b) 1 and 6 (c) 2 and 3 (d) 2 and 4
  28. 28.If a + b = 15 and ab = 50, what is (a − b)²? (a) 125 (b) 5 (c) 225 (d) 25
  29. 29.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 441 (b) 451 (c) 401 (d) 361
  30. 30.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6561 (b) 6401 (c) 6241 (d) 6571
  31. 31.What is the product of the roots of x² − 12x + 35 = 0? (a) 34 (b) 35 (c) 12 (d) 36
  32. 32.If a + b = 15 and ab = 36, what is a² + b²? (a) 297 (b) 225 (c) 189 (d) 153
  33. 33.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 1)(x + 11) (c) (x + 3)(x + 9) (d) (x + 1)(x + 27)
  34. 34.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x + 4)(x + 7) (c) (x + 1)(x + 28) (d) (x − 4)(x − 7)
  35. 35.If a + b = 20 and ab = 96, what is a² + b²? (a) 592 (b) 304 (c) 208 (d) 400
  36. 36.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 3)(x + 4) (c) (x + 1)(x + 6) (d) (x + 1)(x + 12)
  37. 37.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3575 (b) 3025 (c) 4225 (d) 3035
  38. 38.If a + b = 14 and ab = 48, what is (a − b)²? (a) 196 (b) 4 (c) 2 (d) 100
  39. 39.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1225 (b) 925 (c) 1235 (d) 625
  40. 40.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 909 (b) 1099 (c) 1089 (d) 729

Answer key

  1. 2116
  2. 7921
  3. 4 and 5
  4. 136
  5. 7 and 8
  6. 18
  7. 225
  8. 1
  9. 34
  10. 16
  11. 145
  12. 15
  13. 8 and 9
  14. 185
  15. 9
  16. 4225
  17. 8
  18. 1
  19. 193
  20. 137
  21. 5
  22. 149
  23. (x + 4)(x + 9)
  24. 36
  25. 181
  26. 841
  27. 2 and 3
  28. 25
  29. 441
  30. 6561
  31. 35
  32. 153
  33. (x + 3)(x + 9)
  34. (x + 4)(x + 7)
  35. 208
  36. (x + 3)(x + 4)
  37. 3025
  38. 4
  39. 1225
  40. 1089

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