← Identities, polynomials and quadratic roots lesson New set →

Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 916 · 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 = 9, what is (a − b)²? (a) 8 (b) 82 (c) 64 (d) 100
  2. 2.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1575 (b) 1235 (c) 2025 (d) 1225
  3. 3.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 16) (c) (x − 8)(x − 9) (d) (x + 1)(x + 72)
  4. 4.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 11) (b) (x − 4)(x − 8) (c) (x + 4)(x + 8) (d) (x + 1)(x + 32)
  5. 5.What is the sum of the roots of x² − 8x + 12 = 0? (a) 9 (b) 8 (c) 12 (d) 7
  6. 6.If a + b = 10 and ab = 16, what is (a − b)²? (a) 100 (b) 6 (c) 36 (d) 68
  7. 7.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4875 (b) 4235 (c) 5625 (d) 4225
  8. 8.Which is the correct factorisation of x² + 7x + 12? (a) (x + 3)(x + 4) (b) (x − 3)(x − 4) (c) (x + 1)(x + 12) (d) (x + 1)(x + 6)
  9. 9.What is the sum of the roots of x² − 10x + 21 = 0? (a) 21 (b) 11 (c) 10 (d) 9
  10. 10.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x − 7)(x − 9) (c) (x + 1)(x + 15) (d) (x + 7)(x + 9)
  11. 11.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 2 and 36 (c) 1 and 16 (d) 8 and 9
  12. 12.If a + b = 11 and ab = 24, what is a² + b²? (a) 121 (b) 169 (c) 97 (d) 73
  13. 13.What are the roots of x² − 13x + 40 = 0? (a) 2 and 20 (b) 5 and 8 (c) 1 and 40 (d) 1 and 12
  14. 14.What are the roots of x² − 5x + 6 = 0? (a) 1 and 6 (b) 2 and 3 (c) 2 and 4 (d) 1 and 4
  15. 15.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 400 (c) 202 (d) 4
  16. 16.If a + b = 5 and ab = 4, what is a² + b²? (a) 25 (b) 33 (c) 21 (d) 17
  17. 17.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 2)(x + 4) (c) (x + 1)(x + 5) (d) (x − 2)(x − 4)
  18. 18.If a + b = 17 and ab = 72, what is a² + b²? (a) 145 (b) 433 (c) 217 (d) 289
  19. 19.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 14) (c) (x + 1)(x + 56) (d) (x − 7)(x − 8)
  20. 20.If a + b = 13 and ab = 36, what is a² + b²? (a) 133 (b) 169 (c) 241 (d) 97
  21. 21.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1521 (b) 1691 (c) 1681 (d) 1601
  22. 22.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 3)(x + 6) (c) (x + 1)(x + 18) (d) (x + 1)(x + 8)
  23. 23.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 1)(x + 6) (c) (x + 1)(x + 4) (d) (x + 2)(x + 3)
  24. 24.If a + b = 11 and ab = 28, what is (a − b)²? (a) 9 (b) 3 (c) 121 (d) 65
  25. 25.If a + b = 11 and ab = 30, what is (a − b)²? (a) 1 (b) 91 (c) 61 (d) 121
  26. 26.What are the roots of x² − 15x + 56 = 0? (a) 7 and 8 (b) 1 and 14 (c) 2 and 28 (d) 1 and 56
  27. 27.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1099 (b) 909 (c) 1089 (d) 729
  28. 28.If a + b = 8 and ab = 7, what is a² + b²? (a) 64 (b) 78 (c) 50 (d) 57
  29. 29.If a + b = 11 and ab = 28, what is a² + b²? (a) 93 (b) 121 (c) 177 (d) 65
  30. 30.What is the sum of the roots of x² − 13x + 36 = 0? (a) 14 (b) 36 (c) 12 (d) 13
  31. 31.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x + 1)(x + 7) (c) (x + 3)(x + 5) (d) (x − 3)(x − 5)
  32. 32.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x + 1)(x + 13) (c) (x + 5)(x + 9) (d) (x − 5)(x − 9)
  33. 33.What is the sum of the roots of x² − 9x + 14 = 0? (a) 8 (b) 10 (c) 9 (d) 14
  34. 34.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)
  35. 35.If a + b = 15 and ab = 44, what is a² + b²? (a) 313 (b) 225 (c) 137 (d) 181
  36. 36.What are the roots of x² − 10x + 16 = 0? (a) 3 and 7 (b) 1 and 16 (c) 2 and 8 (d) 1 and 9
  37. 37.What is the product of the roots of x² − 10x + 16 = 0? (a) 17 (b) 16 (c) 10 (d) 15
  38. 38.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x − 4)(x − 9) (c) (x + 1)(x + 12) (d) (x + 4)(x + 9)
  39. 39.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3596 (b) 3364 (c) 3374 (d) 3844
  40. 40.If a + b = 20 and ab = 96, what is (a − b)²? (a) 4 (b) 208 (c) 400 (d) 16

Answer key

  1. 64
  2. 1225
  3. (x + 8)(x + 9)
  4. (x + 4)(x + 8)
  5. 8
  6. 36
  7. 4225
  8. (x + 3)(x + 4)
  9. 10
  10. (x + 7)(x + 9)
  11. 8 and 9
  12. 73
  13. 5 and 8
  14. 2 and 3
  15. 4
  16. 17
  17. (x + 2)(x + 4)
  18. 145
  19. (x + 7)(x + 8)
  20. 97
  21. 1681
  22. (x + 3)(x + 6)
  23. (x + 2)(x + 3)
  24. 9
  25. 1
  26. 7 and 8
  27. 1089
  28. 50
  29. 65
  30. 13
  31. (x + 3)(x + 5)
  32. (x + 5)(x + 9)
  33. 9
  34. (x + 5)(x + 8)
  35. 137
  36. 2 and 8
  37. 16
  38. (x + 4)(x + 9)
  39. 3364
  40. 16

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