← Identities, polynomials and quadratic roots lesson New set →

Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 652 · 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.What is the sum of the roots of x² − 11x + 24 = 0? (a) 12 (b) 24 (c) 11 (d) 10
  2. 2.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2499 (b) 2601 (c) 2401 (d) 2411
  3. 3.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 1)(x + 72) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)
  4. 4.What is the sum of the roots of x² − 9x + 20 = 0? (a) 9 (b) 8 (c) 20 (d) 10
  5. 5.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 14 (c) 16 (d) 15
  6. 6.If a + b = 14 and ab = 45, what is a² + b²? (a) 151 (b) 196 (c) 286 (d) 106
  7. 7.What is the product of the roots of x² − 8x + 12 = 0? (a) 13 (b) 8 (c) 11 (d) 12
  8. 8.What is the product of the roots of x² − 5x + 6 = 0? (a) 5 (b) 8 (c) 6 (d) 7
  9. 9.What is the sum of the roots of x² − 8x + 12 = 0? (a) 7 (b) 12 (c) 9 (d) 8
  10. 10.What is the product of the roots of x² − 12x + 27 = 0? (a) 12 (b) 26 (c) 28 (d) 27
  11. 11.If a + b = 16 and ab = 55, what is a² + b²? (a) 256 (b) 146 (c) 366 (d) 201
  12. 12.If a + b = 16 and ab = 63, what is (a − b)²? (a) 4 (b) 256 (c) 130 (d) 2
  13. 13.If a + b = 14 and ab = 48, what is (a − b)²? (a) 4 (b) 100 (c) 2 (d) 196
  14. 14.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 1)(x + 42) (c) (x − 6)(x − 7) (d) (x + 6)(x + 7)
  15. 15.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 400 (c) 301 (d) 202
  16. 16.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 586 (b) 576 (c) 256 (d) 416
  17. 17.What is the product of the roots of x² − 7x + 10 = 0? (a) 7 (b) 10 (c) 11 (d) 9
  18. 18.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10001 (b) 10211 (c) 10201 (d) 9801
  19. 19.If a + b = 12 and ab = 11, what is a² + b²? (a) 166 (b) 144 (c) 133 (d) 122
  20. 20.If a + b = 13 and ab = 22, what is (a − b)²? (a) 125 (b) 169 (c) 81 (d) 9
  21. 21.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 13) (c) (x + 6)(x + 8) (d) (x + 1)(x + 48)
  22. 22.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 841 (b) 851 (c) 961 (d) 899
  23. 23.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 361 (c) 181 (d) 541
  24. 24.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 784 (b) 904 (c) 1024 (d) 1034
  25. 25.What are the roots of x² − 15x + 56 = 0? (a) 2 and 28 (b) 7 and 8 (c) 1 and 14 (d) 1 and 56
  26. 26.What is the sum of the roots of x² − 12x + 27 = 0? (a) 27 (b) 13 (c) 11 (d) 12
  27. 27.If a + b = 9 and ab = 8, what is (a − b)²? (a) 81 (b) 65 (c) 7 (d) 49
  28. 28.If a + b = 12 and ab = 27, what is (a − b)²? (a) 144 (b) 36 (c) 90 (d) 6
  29. 29.If a + b = 18 and ab = 77, what is (a − b)²? (a) 170 (b) 16 (c) 324 (d) 4
  30. 30.If a + b = 11 and ab = 28, what is a² + b²? (a) 65 (b) 177 (c) 121 (d) 93
  31. 31.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1859 (b) 1369 (c) 1609 (d) 1849
  32. 32.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4896 (b) 4624 (c) 5184 (d) 4634
  33. 33.What is the sum of the roots of x² − 6x + 8 = 0? (a) 5 (b) 6 (c) 8 (d) 7
  34. 34.What is the product of the roots of x² − 13x + 36 = 0? (a) 35 (b) 13 (c) 37 (d) 36
  35. 35.What is the sum of the roots of x² − 12x + 32 = 0? (a) 11 (b) 13 (c) 12 (d) 32
  36. 36.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 73 (c) 71 (d) 72
  37. 37.If a + b = 11 and ab = 30, what is a² + b²? (a) 91 (b) 181 (c) 61 (d) 121
  38. 38.If a + b = 9 and ab = 20, what is a² + b²? (a) 81 (b) 121 (c) 41 (d) 61
  39. 39.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 4225 (b) 3035 (c) 3025 (d) 3575
  40. 40.What are the roots of x² − 11x + 18 = 0? (a) 1 and 18 (b) 1 and 10 (c) 3 and 6 (d) 2 and 9

Answer key

  1. 11
  2. 2401
  3. (x + 8)(x + 9)
  4. 9
  5. 15
  6. 106
  7. 12
  8. 6
  9. 8
  10. 27
  11. 146
  12. 4
  13. 4
  14. (x + 6)(x + 7)
  15. 202
  16. 576
  17. 10
  18. 10201
  19. 122
  20. 81
  21. (x + 6)(x + 8)
  22. 841
  23. 181
  24. 1024
  25. 7 and 8
  26. 12
  27. 49
  28. 36
  29. 16
  30. 65
  31. 1849
  32. 4624
  33. 6
  34. 36
  35. 12
  36. 72
  37. 61
  38. 41
  39. 3025
  40. 2 and 9

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