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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 344 · 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 73²? (a) 5339 (b) 4489 (c) 4909 (d) 5329
  2. 2.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 15 (c) 16 (d) 56
  3. 3.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 384 (b) 266 (c) 256 (d) 576
  4. 4.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 2)(x + 4) (c) (x − 2)(x − 4) (d) (x + 1)(x + 5)
  5. 5.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x − 6)(x − 8) (c) (x + 1)(x + 13) (d) (x + 6)(x + 8)
  6. 6.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 72 (c) 17 (d) 73
  7. 7.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 289 (c) 217 (d) 145
  8. 8.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 5635 (b) 6375 (c) 5625 (d) 7225
  9. 9.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 16)
  10. 10.If a + b = 16 and ab = 60, what is (a − b)²? (a) 136 (b) 16 (c) 256 (d) 4
  11. 11.Which is the correct factorisation of x² + 11x + 24? (a) (x + 3)(x + 8) (b) (x − 3)(x − 8) (c) (x + 1)(x + 10) (d) (x + 1)(x + 24)
  12. 12.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x − 5)(x − 6) (c) (x + 1)(x + 30) (d) (x + 1)(x + 10)
  13. 13.What is the product of the roots of x² − 11x + 28 = 0? (a) 29 (b) 11 (c) 28 (d) 27
  14. 14.What are the roots of x² − 10x + 24 = 0? (a) 1 and 9 (b) 4 and 6 (c) 2 and 12 (d) 1 and 24
  15. 15.If a + b = 10 and ab = 21, what is (a − b)²? (a) 100 (b) 58 (c) 4 (d) 16
  16. 16.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x − 7)(x − 8) (c) (x + 7)(x + 8) (d) (x + 1)(x + 56)
  17. 17.What are the roots of x² − 5x + 6 = 0? (a) 2 and 4 (b) 2 and 3 (c) 1 and 6 (d) 1 and 4
  18. 18.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x + 2)(x + 8) (c) (x − 2)(x − 8) (d) (x + 1)(x + 16)
  19. 19.If a + b = 9 and ab = 18, what is (a − b)²? (a) 81 (b) 3 (c) 45 (d) 9
  20. 20.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 6)(x + 7) (c) (x + 1)(x + 42) (d) (x − 6)(x − 7)
  21. 21.What are the roots of x² − 15x + 56 = 0? (a) 1 and 14 (b) 2 and 28 (c) 7 and 8 (d) 1 and 56
  22. 22.If a + b = 20 and ab = 99, what is a² + b²? (a) 202 (b) 301 (c) 400 (d) 598
  23. 23.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8464 (b) 7744 (c) 8104 (d) 8474
  24. 24.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x + 1)(x + 13) (c) (x + 1)(x + 45) (d) (x − 5)(x − 9)
  25. 25.If a + b = 13 and ab = 42, what is (a − b)²? (a) 169 (b) 1 (c) 127 (d) 85
  26. 26.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x + 3)(x + 7) (c) (x − 3)(x − 7) (d) (x + 1)(x + 9)
  27. 27.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3625 (b) 3025 (c) 4225 (d) 4235
  28. 28.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2704 (b) 2504 (c) 2714 (d) 2304
  29. 29.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x − 2)(x − 6) (c) (x + 1)(x + 12) (d) (x + 2)(x + 6)
  30. 30.If a + b = 18 and ab = 80, what is (a − b)²? (a) 2 (b) 324 (c) 164 (d) 4
  31. 31.Which is the correct factorisation of x² + 9x + 18? (a) (x + 3)(x + 6) (b) (x + 1)(x + 8) (c) (x − 3)(x − 6) (d) (x + 1)(x + 18)
  32. 32.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7744 (b) 8464 (c) 8096 (d) 7754
  33. 33.If a + b = 17 and ab = 70, what is (a − b)²? (a) 3 (b) 149 (c) 9 (d) 289
  34. 34.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6561 (b) 6251 (c) 6241 (d) 6399
  35. 35.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 361 (c) 181 (d) 1
  36. 36.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4891 (b) 4499 (c) 5329 (d) 4489
  37. 37.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9991 (b) 9409 (c) 10609 (d) 9419
  38. 38.What is the product of the roots of x² − 5x + 6 = 0? (a) 6 (b) 7 (c) 8 (d) 5
  39. 39.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1369 (b) 1591 (c) 1379 (d) 1849
  40. 40.If a + b = 8 and ab = 15, what is a² + b²? (a) 64 (b) 49 (c) 34 (d) 94

Answer key

  1. 5329
  2. 15
  3. 256
  4. (x + 2)(x + 4)
  5. (x + 6)(x + 8)
  6. 72
  7. 145
  8. 5625
  9. (x + 8)(x + 9)
  10. 16
  11. (x + 3)(x + 8)
  12. (x + 5)(x + 6)
  13. 28
  14. 4 and 6
  15. 16
  16. (x + 7)(x + 8)
  17. 2 and 3
  18. (x + 2)(x + 8)
  19. 9
  20. (x + 6)(x + 7)
  21. 7 and 8
  22. 202
  23. 8464
  24. (x + 5)(x + 9)
  25. 1
  26. (x + 3)(x + 7)
  27. 4225
  28. 2704
  29. (x + 2)(x + 6)
  30. 4
  31. (x + 3)(x + 6)
  32. 7744
  33. 9
  34. 6241
  35. 1
  36. 4489
  37. 9409
  38. 6
  39. 1369
  40. 34

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