← Identities, polynomials and quadratic roots lesson New set →

Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 636 · 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 31²? (a) 901 (b) 971 (c) 841 (d) 961
  2. 2.If a + b = 16 and ab = 48, what is (a − b)²? (a) 160 (b) 8 (c) 64 (d) 256
  3. 3.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)
  4. 4.If a + b = 19 and ab = 84, what is a² + b²? (a) 361 (b) 529 (c) 277 (d) 193
  5. 5.What are the roots of x² − 12x + 35 = 0? (a) 1 and 35 (b) 2 and 10 (c) 1 and 11 (d) 5 and 7
  6. 6.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6561 (b) 6251 (c) 6241 (d) 6399
  7. 7.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 784 (c) 896 (d) 1024
  8. 8.If a + b = 9 and ab = 18, what is (a − b)²? (a) 9 (b) 3 (c) 81 (d) 45
  9. 9.What is the sum of the roots of x² − 9x + 20 = 0? (a) 20 (b) 9 (c) 8 (d) 10
  10. 10.If a + b = 9 and ab = 20, what is (a − b)²? (a) 61 (b) 81 (c) 41 (d) 1
  11. 11.If a + b = 7 and ab = 12, what is a² + b²? (a) 49 (b) 37 (c) 73 (d) 25
  12. 12.What are the roots of x² − 12x + 32 = 0? (a) 4 and 8 (b) 2 and 16 (c) 1 and 11 (d) 1 and 32
  13. 13.If a + b = 7 and ab = 12, what is (a − b)²? (a) 49 (b) 1 (c) 25 (d) 37
  14. 14.If a + b = 21 and ab = 108, what is (a − b)²? (a) 9 (b) 441 (c) 3 (d) 225
  15. 15.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2035 (b) 1225 (c) 2025 (d) 1625
  16. 16.If a + b = 16 and ab = 55, what is (a − b)²? (a) 256 (b) 146 (c) 6 (d) 36
  17. 17.If a + b = 17 and ab = 70, what is a² + b²? (a) 219 (b) 429 (c) 149 (d) 289
  18. 18.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)
  19. 19.What is the sum of the roots of x² − 9x + 18 = 0? (a) 10 (b) 8 (c) 9 (d) 18
  20. 20.What is the product of the roots of x² − 13x + 40 = 0? (a) 41 (b) 39 (c) 13 (d) 40
  21. 21.What are the roots of x² − 8x + 15 = 0? (a) 1 and 7 (b) 2 and 6 (c) 3 and 5 (d) 1 and 15
  22. 22.If a + b = 11 and ab = 30, what is a² + b²? (a) 181 (b) 121 (c) 91 (d) 61
  23. 23.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1379 (b) 1369 (c) 1591 (d) 1849
  24. 24.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1099 (b) 1089 (c) 909 (d) 729
  25. 25.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x − 4)(x − 8) (c) (x + 1)(x + 11) (d) (x + 4)(x + 8)
  26. 26.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1616 (b) 1296 (c) 1936 (d) 1946
  27. 27.What is the sum of the roots of x² − 13x + 42 = 0? (a) 42 (b) 13 (c) 14 (d) 12
  28. 28.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 56) (c) (x + 7)(x + 8) (d) (x + 1)(x + 14)
  29. 29.If a + b = 13 and ab = 36, what is (a − b)²? (a) 5 (b) 169 (c) 25 (d) 97
  30. 30.If a + b = 18 and ab = 72, what is (a − b)²? (a) 324 (b) 36 (c) 6 (d) 180
  31. 31.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 899 (b) 841 (c) 851 (d) 961
  32. 32.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 441 (b) 361 (c) 371 (d) 399
  33. 33.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 441 (b) 361 (c) 401 (d) 451
  34. 34.If a + b = 14 and ab = 33, what is (a − b)²? (a) 8 (b) 196 (c) 64 (d) 130
  35. 35.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10004 (b) 9604 (c) 10414 (d) 10404
  36. 36.What is the sum of the roots of x² − 14x + 48 = 0? (a) 14 (b) 48 (c) 15 (d) 13
  37. 37.If a + b = 13 and ab = 42, what is a² + b²? (a) 169 (b) 127 (c) 85 (d) 253
  38. 38.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 48) (c) (x + 6)(x + 8) (d) (x + 1)(x + 13)
  39. 39.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x − 2)(x − 6) (c) (x + 1)(x + 7) (d) (x + 2)(x + 6)
  40. 40.If a + b = 17 and ab = 60, what is (a − b)²? (a) 7 (b) 49 (c) 289 (d) 169

Answer key

  1. 961
  2. 64
  3. (x + 3)(x + 7)
  4. 193
  5. 5 and 7
  6. 6241
  7. 784
  8. 9
  9. 9
  10. 1
  11. 25
  12. 4 and 8
  13. 1
  14. 9
  15. 2025
  16. 36
  17. 149
  18. (x + 8)(x + 9)
  19. 9
  20. 40
  21. 3 and 5
  22. 61
  23. 1369
  24. 1089
  25. (x + 4)(x + 8)
  26. 1936
  27. 13
  28. (x + 7)(x + 8)
  29. 25
  30. 36
  31. 841
  32. 361
  33. 441
  34. 64
  35. 10404
  36. 14
  37. 85
  38. (x + 6)(x + 8)
  39. (x + 2)(x + 6)
  40. 49

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