← Identities, polynomials and quadratic roots lesson New set →

Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 885 · 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 62²? (a) 3604 (b) 3364 (c) 3854 (d) 3844
  2. 2.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3481 (b) 3601 (c) 3731 (d) 3721
  3. 3.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 1)(x + 10) (c) (x + 5)(x + 6) (d) (x − 5)(x − 6)
  4. 4.What is the product of the roots of x² − 15x + 56 = 0? (a) 56 (b) 57 (c) 55 (d) 15
  5. 5.If a + b = 13 and ab = 30, what is a² + b²? (a) 139 (b) 109 (c) 169 (d) 229
  6. 6.What is the sum of the roots of x² − 6x + 8 = 0? (a) 8 (b) 7 (c) 6 (d) 5
  7. 7.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1369 (b) 1859 (c) 1849 (d) 1609
  8. 8.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x + 7)(x + 8) (d) (x − 7)(x − 8)
  9. 9.If a + b = 10 and ab = 21, what is (a − b)²? (a) 58 (b) 16 (c) 4 (d) 100
  10. 10.If a + b = 19 and ab = 84, what is a² + b²? (a) 193 (b) 361 (c) 529 (d) 277
  11. 11.If a + b = 16 and ab = 63, what is (a − b)²? (a) 2 (b) 256 (c) 4 (d) 130
  12. 12.If a + b = 10 and ab = 16, what is a² + b²? (a) 84 (b) 100 (c) 68 (d) 132
  13. 13.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 5776 (b) 7056 (c) 5786 (d) 6384
  14. 14.If a + b = 18 and ab = 80, what is a² + b²? (a) 244 (b) 324 (c) 484 (d) 164
  15. 15.If a + b = 20 and ab = 96, what is a² + b²? (a) 592 (b) 304 (c) 208 (d) 400
  16. 16.If a + b = 16 and ab = 55, what is a² + b²? (a) 366 (b) 256 (c) 146 (d) 201
  17. 17.If a + b = 17 and ab = 72, what is a² + b²? (a) 289 (b) 217 (c) 433 (d) 145
  18. 18.If a + b = 17 and ab = 70, what is a² + b²? (a) 429 (b) 149 (c) 219 (d) 289
  19. 19.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 5041 (b) 4899 (c) 4761 (d) 4771
  20. 20.If a + b = 17 and ab = 66, what is (a − b)²? (a) 25 (b) 289 (c) 157 (d) 5
  21. 21.If a + b = 7 and ab = 12, what is a² + b²? (a) 49 (b) 73 (c) 37 (d) 25
  22. 22.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 12) (c) (x + 3)(x + 4) (d) (x + 1)(x + 6)
  23. 23.If a + b = 19 and ab = 88, what is (a − b)²? (a) 185 (b) 9 (c) 3 (d) 361
  24. 24.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 1)(x + 20) (c) (x + 4)(x + 5) (d) (x + 1)(x + 8)
  25. 25.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1024 (b) 904 (c) 784 (d) 1034
  26. 26.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 16 (c) 63 (d) 62
  27. 27.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 541 (c) 181 (d) 361
  28. 28.If a + b = 11 and ab = 18, what is (a − b)²? (a) 7 (b) 121 (c) 49 (d) 85
  29. 29.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)
  30. 30.If a + b = 14 and ab = 24, what is (a − b)²? (a) 148 (b) 100 (c) 196 (d) 10
  31. 31.If a + b = 12 and ab = 27, what is a² + b²? (a) 117 (b) 90 (c) 144 (d) 198
  32. 32.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 72 (c) 73 (d) 17
  33. 33.If a + b = 13 and ab = 36, what is a² + b²? (a) 169 (b) 97 (c) 241 (d) 133
  34. 34.If a + b = 16 and ab = 48, what is a² + b²? (a) 256 (b) 352 (c) 208 (d) 160
  35. 35.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 72 (c) 17 (d) 16
  36. 36.What are the roots of x² − 7x + 10 = 0? (a) 1 and 6 (b) 3 and 4 (c) 1 and 10 (d) 2 and 5
  37. 37.What is the sum of the roots of x² − 9x + 20 = 0? (a) 20 (b) 8 (c) 9 (d) 10
  38. 38.What are the roots of x² − 12x + 35 = 0? (a) 2 and 10 (b) 1 and 11 (c) 5 and 7 (d) 1 and 35
  39. 39.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4489 (b) 5329 (c) 4499 (d) 4891
  40. 40.What is the product of the roots of x² − 14x + 48 = 0? (a) 47 (b) 48 (c) 14 (d) 49

Answer key

  1. 3844
  2. 3721
  3. (x + 5)(x + 6)
  4. 56
  5. 109
  6. 6
  7. 1849
  8. (x + 7)(x + 8)
  9. 16
  10. 193
  11. 4
  12. 68
  13. 5776
  14. 164
  15. 208
  16. 146
  17. 145
  18. 149
  19. 4761
  20. 25
  21. 25
  22. (x + 3)(x + 4)
  23. 9
  24. (x + 4)(x + 5)
  25. 1024
  26. 63
  27. 181
  28. 49
  29. (x + 8)(x + 9)
  30. 100
  31. 90
  32. 72
  33. 97
  34. 160
  35. 17
  36. 2 and 5
  37. 9
  38. 5 and 7
  39. 4489
  40. 48

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