← Identities, polynomials and quadratic roots lesson New set →

Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 845 · 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 = 17 and ab = 72, what is a² + b²? (a) 433 (b) 289 (c) 217 (d) 145
  2. 2.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)
  3. 3.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1369 (b) 1859 (c) 1849 (d) 1609
  4. 4.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 14) (c) (x + 7)(x + 8) (d) (x + 1)(x + 56)
  5. 5.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 16 (c) 17 (d) 18
  6. 6.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x − 7)(x − 9) (c) (x + 1)(x + 63) (d) (x + 1)(x + 15)
  7. 7.If a + b = 10 and ab = 21, what is (a − b)²? (a) 58 (b) 100 (c) 16 (d) 4
  8. 8.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 17 (c) 16 (d) 15
  9. 9.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x + 1)(x + 21) (c) (x + 3)(x + 7) (d) (x − 3)(x − 7)
  10. 10.What are the roots of x² − 13x + 36 = 0? (a) 1 and 12 (b) 4 and 9 (c) 2 and 18 (d) 1 and 36
  11. 11.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 7 and 8 (c) 1 and 14 (d) 2 and 28
  12. 12.Which is the correct factorisation of x² + 7x + 10? (a) (x − 2)(x − 5) (b) (x + 1)(x + 6) (c) (x + 2)(x + 5) (d) (x + 1)(x + 10)
  13. 13.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3364 (b) 3854 (c) 3604 (d) 3844
  14. 14.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 625 (b) 375 (c) 225 (d) 235
  15. 15.What is the product of the roots of x² − 13x + 42 = 0? (a) 13 (b) 41 (c) 43 (d) 42
  16. 16.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1625 (b) 2025 (c) 1225 (d) 2035
  17. 17.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
  18. 18.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 9) (b) (x + 4)(x + 6) (c) (x − 4)(x − 6) (d) (x + 1)(x + 24)
  19. 19.What are the roots of x² − 15x + 54 = 0? (a) 6 and 9 (b) 1 and 14 (c) 1 and 54 (d) 2 and 27
  20. 20.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x + 1)(x + 13) (c) (x + 6)(x + 8) (d) (x − 6)(x − 8)
  21. 21.What is the sum of the roots of x² − 13x + 36 = 0? (a) 12 (b) 36 (c) 14 (d) 13
  22. 22.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10826 (b) 10816 (c) 10016 (d) 9216
  23. 23.What are the roots of x² − 7x + 10 = 0? (a) 2 and 5 (b) 3 and 4 (c) 1 and 6 (d) 1 and 10
  24. 24.If a + b = 15 and ab = 44, what is (a − b)²? (a) 7 (b) 137 (c) 225 (d) 49
  25. 25.If a + b = 19 and ab = 88, what is a² + b²? (a) 537 (b) 185 (c) 273 (d) 361
  26. 26.If a + b = 11 and ab = 28, what is a² + b²? (a) 93 (b) 65 (c) 121 (d) 177
  27. 27.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 5339 (b) 4909 (c) 4489 (d) 5329
  28. 28.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 271 (c) 361 (d) 181
  29. 29.If a + b = 18 and ab = 77, what is (a − b)²? (a) 16 (b) 170 (c) 4 (d) 324
  30. 30.If a + b = 12 and ab = 27, what is (a − b)²? (a) 144 (b) 90 (c) 6 (d) 36
  31. 31.What is the sum of the roots of x² − 11x + 28 = 0? (a) 12 (b) 11 (c) 10 (d) 28
  32. 32.If a + b = 12 and ab = 32, what is (a − b)²? (a) 4 (b) 144 (c) 16 (d) 80
  33. 33.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 15 (c) 14 (d) 56
  34. 34.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1089 (b) 909 (c) 1099 (d) 729
  35. 35.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8091 (b) 8649 (c) 7579 (d) 7569
  36. 36.Which is the correct factorisation of x² + 8x + 15? (a) (x + 3)(x + 5) (b) (x + 1)(x + 15) (c) (x + 1)(x + 7) (d) (x − 3)(x − 5)
  37. 37.If a + b = 19 and ab = 88, what is (a − b)²? (a) 9 (b) 361 (c) 3 (d) 185
  38. 38.What is the product of the roots of x² − 7x + 12 = 0? (a) 12 (b) 13 (c) 11 (d) 7
  39. 39.What is the sum of the roots of x² − 11x + 24 = 0? (a) 11 (b) 10 (c) 24 (d) 12
  40. 40.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9419 (b) 9409 (c) 9991 (d) 10609

Answer key

  1. 145
  2. (x + 8)(x + 9)
  3. 1849
  4. (x + 7)(x + 8)
  5. 17
  6. (x + 7)(x + 9)
  7. 16
  8. 16
  9. (x + 3)(x + 7)
  10. 4 and 9
  11. 7 and 8
  12. (x + 2)(x + 5)
  13. 3844
  14. 225
  15. 42
  16. 2025
  17. 6 and 7
  18. (x + 4)(x + 6)
  19. 6 and 9
  20. (x + 6)(x + 8)
  21. 13
  22. 10816
  23. 2 and 5
  24. 49
  25. 185
  26. 65
  27. 5329
  28. 181
  29. 16
  30. 36
  31. 11
  32. 16
  33. 15
  34. 1089
  35. 7569
  36. (x + 3)(x + 5)
  37. 9
  38. 12
  39. 11
  40. 9409

Free maths olympiad lessons, practice and worksheets at talentjr.in/maths-olympiad