← Identities, polynomials and quadratic roots lesson New set →

Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 241 · 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.If a + b = 10 and ab = 16, what is a² + b²? (a) 100 (b) 132 (c) 68 (d) 84
  2. 2.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
  3. 3.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)
  4. 4.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1369 (b) 1859 (c) 1609 (d) 1849
  5. 5.What is the product of the roots of x² − 14x + 45 = 0? (a) 14 (b) 46 (c) 45 (d) 44
  6. 6.What are the roots of x² − 11x + 30 = 0? (a) 1 and 30 (b) 5 and 6 (c) 1 and 10 (d) 2 and 15
  7. 7.If a + b = 14 and ab = 48, what is (a − b)²? (a) 4 (b) 2 (c) 100 (d) 196
  8. 8.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 3035 (b) 2525 (c) 3025 (d) 2025
  9. 9.If a + b = 12 and ab = 27, what is (a − b)²? (a) 90 (b) 144 (c) 6 (d) 36
  10. 10.What is the product of the roots of x² − 13x + 36 = 0? (a) 36 (b) 35 (c) 13 (d) 37
  11. 11.If a + b = 6 and ab = 8, what is a² + b²? (a) 20 (b) 52 (c) 28 (d) 36
  12. 12.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x + 6)(x + 8) (c) (x − 6)(x − 8) (d) (x + 1)(x + 13)
  13. 13.What are the roots of x² − 9x + 20 = 0? (a) 2 and 10 (b) 4 and 5 (c) 1 and 8 (d) 1 and 20
  14. 14.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 4 (c) 400 (d) 202
  15. 15.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 729 (b) 1089 (c) 1099 (d) 909
  16. 16.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 15 (c) 13 (d) 14
  17. 17.What is the sum of the roots of x² − 11x + 28 = 0? (a) 10 (b) 11 (c) 28 (d) 12
  18. 18.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 904 (b) 784 (c) 1024 (d) 1034
  19. 19.What is the product of the roots of x² − 9x + 20 = 0? (a) 9 (b) 21 (c) 19 (d) 20
  20. 20.If a + b = 9 and ab = 18, what is a² + b²? (a) 81 (b) 117 (c) 45 (d) 63
  21. 21.If a + b = 8 and ab = 12, what is a² + b²? (a) 52 (b) 64 (c) 40 (d) 88
  22. 22.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
  23. 23.Which is the correct factorisation of x² + 9x + 18? (a) (x + 3)(x + 6) (b) (x + 1)(x + 18) (c) (x + 1)(x + 8) (d) (x − 3)(x − 6)
  24. 24.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1444 (b) 1454 (c) 1596 (d) 1764
  25. 25.What are the roots of x² − 7x + 12 = 0? (a) 1 and 6 (b) 3 and 4 (c) 1 and 12 (d) 2 and 6
  26. 26.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x − 5)(x − 8) (c) (x + 5)(x + 8) (d) (x + 1)(x + 40)
  27. 27.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x − 5)(x − 9) (c) (x + 1)(x + 45) (d) (x + 1)(x + 13)
  28. 28.What are the roots of x² − 10x + 21 = 0? (a) 1 and 21 (b) 2 and 8 (c) 3 and 7 (d) 1 and 9
  29. 29.If a + b = 11 and ab = 30, what is (a − b)²? (a) 1 (b) 91 (c) 61 (d) 121
  30. 30.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 916 (b) 1166 (c) 676 (d) 1156
  31. 31.What is the sum of the roots of x² − 9x + 18 = 0? (a) 10 (b) 9 (c) 8 (d) 18
  32. 32.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 15 (c) 17 (d) 63
  33. 33.If a + b = 16 and ab = 60, what is a² + b²? (a) 136 (b) 196 (c) 256 (d) 376
  34. 34.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1521 (b) 1601 (c) 1681 (d) 1691
  35. 35.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 5486 (b) 4356 (c) 5476 (d) 4916
  36. 36.What is the product of the roots of x² − 11x + 24 = 0? (a) 23 (b) 25 (c) 11 (d) 24
  37. 37.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x − 4)(x − 5) (c) (x + 4)(x + 5) (d) (x + 1)(x + 20)
  38. 38.What is the sum of the roots of x² − 6x + 8 = 0? (a) 5 (b) 7 (c) 6 (d) 8
  39. 39.If a + b = 10 and ab = 9, what is a² + b²? (a) 118 (b) 82 (c) 100 (d) 91
  40. 40.What is the product of the roots of x² − 7x + 10 = 0? (a) 7 (b) 10 (c) 9 (d) 11

Answer key

  1. 68
  2. 5 and 7
  3. (x + 7)(x + 8)
  4. 1849
  5. 45
  6. 5 and 6
  7. 4
  8. 3025
  9. 36
  10. 36
  11. 20
  12. (x + 6)(x + 8)
  13. 4 and 5
  14. 4
  15. 1089
  16. 14
  17. 11
  18. 1024
  19. 20
  20. 45
  21. 40
  22. (x + 8)(x + 9)
  23. (x + 3)(x + 6)
  24. 1444
  25. 3 and 4
  26. (x + 5)(x + 8)
  27. (x + 5)(x + 9)
  28. 3 and 7
  29. 1
  30. 1156
  31. 9
  32. 16
  33. 136
  34. 1681
  35. 5476
  36. 24
  37. (x + 4)(x + 5)
  38. 6
  39. 82
  40. 10

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