← Identities, polynomials and quadratic roots lesson New set →

Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 273 · 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 87²? (a) 8649 (b) 7579 (c) 8091 (d) 7569
  2. 2.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 333 (c) 225 (d) 657
  3. 3.What is the sum of the roots of x² − 7x + 10 = 0? (a) 8 (b) 10 (c) 6 (d) 7
  4. 4.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 729 (b) 891 (c) 1089 (d) 739
  5. 5.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 202 (c) 4 (d) 400
  6. 6.What is the sum of the roots of x² − 8x + 12 = 0? (a) 12 (b) 7 (c) 9 (d) 8
  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.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x − 3)(x − 4) (c) (x + 3)(x + 4) (d) (x + 1)(x + 12)
  9. 9.What are the roots of x² − 13x + 42 = 0? (a) 1 and 42 (b) 1 and 12 (c) 2 and 21 (d) 6 and 7
  10. 10.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)
  11. 11.What is the product of the roots of x² − 14x + 45 = 0? (a) 45 (b) 46 (c) 14 (d) 44
  12. 12.If a + b = 16 and ab = 55, what is a² + b²? (a) 256 (b) 146 (c) 201 (d) 366
  13. 13.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 441 (b) 399 (c) 371 (d) 361
  14. 14.If a + b = 13 and ab = 22, what is a² + b²? (a) 169 (b) 125 (c) 147 (d) 213
  15. 15.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x − 7)(x − 8) (c) (x + 1)(x + 14) (d) (x + 1)(x + 56)
  16. 16.If a + b = 5 and ab = 4, what is a² + b²? (a) 33 (b) 17 (c) 25 (d) 21
  17. 17.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
  18. 18.What are the roots of x² − 11x + 30 = 0? (a) 2 and 15 (b) 1 and 10 (c) 1 and 30 (d) 5 and 6
  19. 19.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x + 1)(x + 16) (c) (x − 2)(x − 8) (d) (x + 2)(x + 8)
  20. 20.Which is the correct factorisation of x² + 11x + 24? (a) (x − 3)(x − 8) (b) (x + 1)(x + 24) (c) (x + 1)(x + 10) (d) (x + 3)(x + 8)
  21. 21.If a + b = 15 and ab = 44, what is a² + b²? (a) 181 (b) 225 (c) 137 (d) 313
  22. 22.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1369 (b) 1849 (c) 1859 (d) 1609
  23. 23.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3364 (b) 3844 (c) 3374 (d) 3596
  24. 24.What is the product of the roots of x² − 10x + 21 = 0? (a) 22 (b) 20 (c) 21 (d) 10
  25. 25.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4884 (b) 4366 (c) 4356 (d) 5476
  26. 26.If a + b = 13 and ab = 12, what is (a − b)²? (a) 169 (b) 145 (c) 11 (d) 121
  27. 27.If a + b = 19 and ab = 84, what is (a − b)²? (a) 361 (b) 5 (c) 193 (d) 25
  28. 28.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x − 6)(x − 7) (c) (x + 1)(x + 12) (d) (x + 1)(x + 42)
  29. 29.If a + b = 15 and ab = 50, what is a² + b²? (a) 325 (b) 125 (c) 225 (d) 175
  30. 30.If a + b = 20 and ab = 99, what is a² + b²? (a) 400 (b) 202 (c) 301 (d) 598
  31. 31.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8659 (b) 8649 (c) 7569 (d) 8109
  32. 32.What is the product of the roots of x² − 16x + 63 = 0? (a) 16 (b) 63 (c) 62 (d) 64
  33. 33.What are the roots of x² − 10x + 16 = 0? (a) 1 and 16 (b) 1 and 9 (c) 2 and 8 (d) 3 and 7
  34. 34.What are the roots of x² − 12x + 35 = 0? (a) 1 and 11 (b) 1 and 35 (c) 5 and 7 (d) 2 and 10
  35. 35.If a + b = 9 and ab = 14, what is a² + b²? (a) 81 (b) 67 (c) 109 (d) 53
  36. 36.What is the sum of the roots of x² − 5x + 6 = 0? (a) 4 (b) 7 (c) 6 (d) 5
  37. 37.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 63 (c) 17 (d) 16
  38. 38.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)
  39. 39.What are the roots of x² − 13x + 40 = 0? (a) 5 and 8 (b) 2 and 20 (c) 1 and 12 (d) 1 and 40
  40. 40.If a + b = 13 and ab = 30, what is a² + b²? (a) 139 (b) 169 (c) 229 (d) 109

Answer key

  1. 7569
  2. 225
  3. 7
  4. 729
  5. 4
  6. 8
  7. (x + 5)(x + 7)
  8. (x + 3)(x + 4)
  9. 6 and 7
  10. (x + 2)(x + 5)
  11. 45
  12. 146
  13. 361
  14. 125
  15. (x + 7)(x + 8)
  16. 17
  17. 8 and 9
  18. 5 and 6
  19. (x + 2)(x + 8)
  20. (x + 3)(x + 8)
  21. 137
  22. 1849
  23. 3364
  24. 21
  25. 4356
  26. 121
  27. 25
  28. (x + 6)(x + 7)
  29. 125
  30. 202
  31. 8649
  32. 63
  33. 2 and 8
  34. 5 and 7
  35. 53
  36. 5
  37. 16
  38. (x + 8)(x + 9)
  39. 5 and 8
  40. 109

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