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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 261 · 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.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x − 2)(x − 9) (c) (x + 1)(x + 18) (d) (x + 2)(x + 9)
  2. 2.What is the product of the roots of x² − 14x + 48 = 0? (a) 48 (b) 14 (c) 49 (d) 47
  3. 3.What is the product of the roots of x² − 13x + 42 = 0? (a) 42 (b) 43 (c) 13 (d) 41
  4. 4.If a + b = 9 and ab = 14, what is a² + b²? (a) 109 (b) 67 (c) 53 (d) 81
  5. 5.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1225 (b) 925 (c) 1235 (d) 625
  6. 6.What are the roots of x² − 13x + 40 = 0? (a) 5 and 8 (b) 1 and 40 (c) 2 and 20 (d) 1 and 12
  7. 7.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x − 5)(x − 7) (c) (x + 5)(x + 7) (d) (x + 1)(x + 11)
  8. 8.What is the sum of the roots of x² − 13x + 40 = 0? (a) 13 (b) 40 (c) 14 (d) 12
  9. 9.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 6375 (b) 5635 (c) 5625 (d) 7225
  10. 10.If a + b = 15 and ab = 56, what is a² + b²? (a) 225 (b) 337 (c) 169 (d) 113
  11. 11.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x + 1)(x + 45) (c) (x + 5)(x + 9) (d) (x − 5)(x − 9)
  12. 12.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 361 (b) 441 (c) 399 (d) 371
  13. 13.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x + 1)(x + 9) (c) (x − 4)(x − 6) (d) (x + 4)(x + 6)
  14. 14.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
  15. 15.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)
  16. 16.If a + b = 13 and ab = 22, what is (a − b)²? (a) 81 (b) 169 (c) 9 (d) 125
  17. 17.If a + b = 17 and ab = 72, what is a² + b²? (a) 145 (b) 289 (c) 433 (d) 217
  18. 18.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1946 (b) 1616 (c) 1936 (d) 1296
  19. 19.What is the product of the roots of x² − 13x + 40 = 0? (a) 41 (b) 39 (c) 13 (d) 40
  20. 20.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 784 (b) 1024 (c) 904 (d) 1034
  21. 21.If a + b = 20 and ab = 99, what is a² + b²? (a) 400 (b) 598 (c) 301 (d) 202
  22. 22.If a + b = 21 and ab = 108, what is (a − b)²? (a) 3 (b) 441 (c) 225 (d) 9
  23. 23.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3854 (b) 3364 (c) 3844 (d) 3604
  24. 24.What are the roots of x² − 13x + 42 = 0? (a) 1 and 42 (b) 2 and 21 (c) 6 and 7 (d) 1 and 12
  25. 25.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x + 1)(x + 12) (c) (x − 6)(x − 7) (d) (x + 1)(x + 42)
  26. 26.What is the product of the roots of x² − 15x + 54 = 0? (a) 55 (b) 15 (c) 53 (d) 54
  27. 27.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 7056 (b) 6384 (c) 5786 (d) 5776
  28. 28.If a + b = 16 and ab = 60, what is (a − b)²? (a) 4 (b) 256 (c) 136 (d) 16
  29. 29.What are the roots of x² − 6x + 8 = 0? (a) 1 and 8 (b) 2 and 5 (c) 1 and 5 (d) 2 and 4
  30. 30.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x − 5)(x − 8) (c) (x + 5)(x + 8) (d) (x + 1)(x + 40)
  31. 31.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2499 (b) 2601 (c) 2411 (d) 2401
  32. 32.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x − 3)(x − 4) (c) (x + 1)(x + 12) (d) (x + 3)(x + 4)
  33. 33.What are the roots of x² − 9x + 18 = 0? (a) 3 and 6 (b) 1 and 8 (c) 2 and 9 (d) 1 and 18
  34. 34.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1444 (b) 1454 (c) 1764 (d) 1596
  35. 35.What are the roots of x² − 11x + 28 = 0? (a) 1 and 28 (b) 4 and 7 (c) 1 and 10 (d) 2 and 14
  36. 36.What are the roots of x² − 9x + 14 = 0? (a) 3 and 6 (b) 2 and 7 (c) 1 and 8 (d) 1 and 14
  37. 37.If a + b = 16 and ab = 63, what is a² + b²? (a) 130 (b) 256 (c) 193 (d) 382
  38. 38.If a + b = 13 and ab = 22, what is a² + b²? (a) 213 (b) 125 (c) 169 (d) 147
  39. 39.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x − 2)(x − 4) (c) (x + 2)(x + 4) (d) (x + 1)(x + 8)
  40. 40.What is the sum of the roots of x² − 11x + 24 = 0? (a) 10 (b) 24 (c) 12 (d) 11

Answer key

  1. (x + 2)(x + 9)
  2. 48
  3. 42
  4. 53
  5. 1225
  6. 5 and 8
  7. (x + 5)(x + 7)
  8. 13
  9. 5625
  10. 113
  11. (x + 5)(x + 9)
  12. 361
  13. (x + 4)(x + 6)
  14. (x + 8)(x + 9)
  15. (x + 7)(x + 8)
  16. 81
  17. 145
  18. 1936
  19. 40
  20. 1024
  21. 202
  22. 9
  23. 3844
  24. 6 and 7
  25. (x + 6)(x + 7)
  26. 54
  27. 5776
  28. 16
  29. 2 and 4
  30. (x + 5)(x + 8)
  31. 2401
  32. (x + 3)(x + 4)
  33. 3 and 6
  34. 1444
  35. 4 and 7
  36. 2 and 7
  37. 130
  38. 125
  39. (x + 2)(x + 4)
  40. 11

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