← Identities, polynomials and quadratic roots lesson New set →

Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 798 · 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 55²? (a) 3575 (b) 3035 (c) 3025 (d) 4225
  2. 2.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4366 (b) 4356 (c) 4884 (d) 5476
  3. 3.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x + 5)(x + 7) (c) (x + 1)(x + 11) (d) (x − 5)(x − 7)
  4. 4.If a + b = 17 and ab = 72, what is a² + b²? (a) 145 (b) 433 (c) 289 (d) 217
  5. 5.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 4)(x + 6) (c) (x + 1)(x + 9) (d) (x + 1)(x + 24)
  6. 6.What is the product of the roots of x² − 9x + 18 = 0? (a) 17 (b) 18 (c) 9 (d) 19
  7. 7.If a + b = 17 and ab = 70, what is a² + b²? (a) 429 (b) 219 (c) 149 (d) 289
  8. 8.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1859 (b) 1369 (c) 1609 (d) 1849
  9. 9.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 15) (c) (x − 7)(x − 9) (d) (x + 1)(x + 63)
  10. 10.What are the roots of x² − 13x + 40 = 0? (a) 1 and 40 (b) 1 and 12 (c) 5 and 8 (d) 2 and 20
  11. 11.If a + b = 9 and ab = 18, what is a² + b²? (a) 45 (b) 117 (c) 81 (d) 63
  12. 12.If a + b = 16 and ab = 48, what is a² + b²? (a) 352 (b) 256 (c) 160 (d) 208
  13. 13.If a + b = 6 and ab = 8, what is (a − b)²? (a) 36 (b) 20 (c) 2 (d) 4
  14. 14.What is the product of the roots of x² − 9x + 20 = 0? (a) 19 (b) 21 (c) 9 (d) 20
  15. 15.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 11025 (b) 11035 (c) 9025 (d) 10025
  16. 16.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3591 (b) 3969 (c) 3249 (d) 3259
  17. 17.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 676 (b) 1166 (c) 916 (d) 1156
  18. 18.If a + b = 11 and ab = 18, what is a² + b²? (a) 103 (b) 121 (c) 85 (d) 157
  19. 19.If a + b = 15 and ab = 36, what is (a − b)²? (a) 153 (b) 225 (c) 9 (d) 81
  20. 20.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6094 (b) 6396 (c) 6724 (d) 6084
  21. 21.If a + b = 9 and ab = 14, what is a² + b²? (a) 53 (b) 109 (c) 67 (d) 81
  22. 22.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 299 (b) 529 (c) 289 (d) 391
  23. 23.If a + b = 16 and ab = 48, what is (a − b)²? (a) 256 (b) 64 (c) 8 (d) 160
  24. 24.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5041 (b) 4901 (c) 4761 (d) 5051
  25. 25.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)
  26. 26.If a + b = 11 and ab = 10, what is (a − b)²? (a) 9 (b) 81 (c) 121 (d) 101
  27. 27.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)
  28. 28.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 2)(x + 3) (c) (x + 1)(x + 4) (d) (x + 1)(x + 6)
  29. 29.Which is the correct factorisation of x² + 8x + 15? (a) (x + 3)(x + 5) (b) (x − 3)(x − 5) (c) (x + 1)(x + 15) (d) (x + 1)(x + 7)
  30. 30.What is the sum of the roots of x² − 12x + 32 = 0? (a) 11 (b) 32 (c) 13 (d) 12
  31. 31.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8101 (b) 7921 (c) 8291 (d) 8281
  32. 32.If a + b = 9 and ab = 20, what is (a − b)²? (a) 1 (b) 41 (c) 81 (d) 61
  33. 33.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x + 3)(x + 7) (c) (x + 1)(x + 21) (d) (x − 3)(x − 7)
  34. 34.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 17 (c) 72 (d) 71
  35. 35.If a + b = 10 and ab = 24, what is a² + b²? (a) 100 (b) 148 (c) 76 (d) 52
  36. 36.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)
  37. 37.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 32) (c) (x + 4)(x + 8) (d) (x + 1)(x + 11)
  38. 38.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3616 (b) 3136 (c) 4106 (d) 4096
  39. 39.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1024 (b) 904 (c) 784 (d) 1034
  40. 40.What is the product of the roots of x² − 16x + 63 = 0? (a) 63 (b) 64 (c) 62 (d) 16

Answer key

  1. 3025
  2. 4356
  3. (x + 5)(x + 7)
  4. 145
  5. (x + 4)(x + 6)
  6. 18
  7. 149
  8. 1849
  9. (x + 7)(x + 9)
  10. 5 and 8
  11. 45
  12. 160
  13. 4
  14. 20
  15. 11025
  16. 3249
  17. 1156
  18. 85
  19. 81
  20. 6084
  21. 53
  22. 289
  23. 64
  24. 5041
  25. (x + 8)(x + 9)
  26. 81
  27. (x + 7)(x + 8)
  28. (x + 2)(x + 3)
  29. (x + 3)(x + 5)
  30. 12
  31. 8281
  32. 1
  33. (x + 3)(x + 7)
  34. 72
  35. 52
  36. (x + 6)(x + 7)
  37. (x + 4)(x + 8)
  38. 4096
  39. 1024
  40. 63

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