← Identities, polynomials and quadratic roots lesson New set →

Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 150 · 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 − 8x + 12 = 0? (a) 7 (b) 12 (c) 9 (d) 8
  2. 2.If a + b = 18 and ab = 80, what is (a − b)²? (a) 164 (b) 2 (c) 4 (d) 324
  3. 3.If a + b = 11 and ab = 30, what is (a − b)²? (a) 91 (b) 61 (c) 121 (d) 1
  4. 4.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6241 (b) 6561 (c) 6401 (d) 6571
  5. 5.What is the sum of the roots of x² − 13x + 40 = 0? (a) 40 (b) 14 (c) 12 (d) 13
  6. 6.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x + 1)(x + 12) (c) (x + 2)(x + 6) (d) (x − 2)(x − 6)
  7. 7.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x − 6)(x − 7) (c) (x + 6)(x + 7) (d) (x + 1)(x + 12)
  8. 8.If a + b = 14 and ab = 45, what is (a − b)²? (a) 4 (b) 16 (c) 106 (d) 196
  9. 9.If a + b = 20 and ab = 96, what is a² + b²? (a) 400 (b) 208 (c) 592 (d) 304
  10. 10.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)
  11. 11.If a + b = 9 and ab = 20, what is a² + b²? (a) 61 (b) 121 (c) 41 (d) 81
  12. 12.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1599 (b) 1681 (c) 1531 (d) 1521
  13. 13.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 18 (c) 16 (d) 72
  14. 14.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8649 (b) 7569 (c) 8659 (d) 8109
  15. 15.If a + b = 16 and ab = 48, what is a² + b²? (a) 352 (b) 256 (c) 160 (d) 208
  16. 16.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3969 (b) 3609 (c) 3249 (d) 3979
  17. 17.Which is the correct factorisation of x² + 11x + 24? (a) (x + 3)(x + 8) (b) (x + 1)(x + 24) (c) (x + 1)(x + 10) (d) (x − 3)(x − 8)
  18. 18.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 1)(x + 30) (c) (x − 5)(x − 6) (d) (x + 5)(x + 6)
  19. 19.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 361 (b) 401 (c) 451 (d) 441
  20. 20.If a + b = 13 and ab = 12, what is a² + b²? (a) 145 (b) 157 (c) 193 (d) 169
  21. 21.If a + b = 16 and ab = 48, what is (a − b)²? (a) 256 (b) 160 (c) 64 (d) 8
  22. 22.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 7744 (b) 8104 (c) 8474 (d) 8464
  23. 23.If a + b = 14 and ab = 24, what is a² + b²? (a) 244 (b) 196 (c) 172 (d) 148
  24. 24.If a + b = 4 and ab = 3, what is (a − b)²? (a) 10 (b) 2 (c) 4 (d) 16
  25. 25.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3364 (b) 3596 (c) 3374 (d) 3844
  26. 26.What is the sum of the roots of x² − 10x + 16 = 0? (a) 10 (b) 9 (c) 11 (d) 16
  27. 27.If a + b = 18 and ab = 80, what is a² + b²? (a) 324 (b) 244 (c) 164 (d) 484
  28. 28.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x + 3)(x + 7) (c) (x − 3)(x − 7) (d) (x + 1)(x + 21)
  29. 29.What is the product of the roots of x² − 6x + 8 = 0? (a) 9 (b) 8 (c) 6 (d) 7
  30. 30.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9025 (b) 9035 (c) 11025 (d) 9975
  31. 31.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2116 (b) 2484 (c) 2126 (d) 2916
  32. 32.If a + b = 12 and ab = 11, what is (a − b)²? (a) 122 (b) 100 (c) 144 (d) 10
  33. 33.What is the product of the roots of x² − 9x + 20 = 0? (a) 21 (b) 9 (c) 20 (d) 19
  34. 34.Which is the correct factorisation of x² + 12x + 32? (a) (x + 4)(x + 8) (b) (x + 1)(x + 32) (c) (x + 1)(x + 11) (d) (x − 4)(x − 8)
  35. 35.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 225 (c) 333 (d) 657
  36. 36.What is the sum of the roots of x² − 10x + 21 = 0? (a) 10 (b) 9 (c) 21 (d) 11
  37. 37.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 71 (c) 73 (d) 72
  38. 38.If a + b = 15 and ab = 56, what is (a − b)²? (a) 113 (b) 1 (c) 225 (d) 169
  39. 39.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x + 3)(x + 5) (c) (x + 1)(x + 7) (d) (x − 3)(x − 5)
  40. 40.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6889 (b) 5929 (c) 6391 (d) 5939

Answer key

  1. 8
  2. 4
  3. 1
  4. 6561
  5. 13
  6. (x + 2)(x + 6)
  7. (x + 6)(x + 7)
  8. 16
  9. 208
  10. (x + 2)(x + 4)
  11. 41
  12. 1521
  13. 17
  14. 8649
  15. 160
  16. 3969
  17. (x + 3)(x + 8)
  18. (x + 5)(x + 6)
  19. 441
  20. 145
  21. 64
  22. 8464
  23. 148
  24. 4
  25. 3364
  26. 10
  27. 164
  28. (x + 3)(x + 7)
  29. 8
  30. 9025
  31. 2116
  32. 100
  33. 20
  34. (x + 4)(x + 8)
  35. 225
  36. 10
  37. 72
  38. 1
  39. (x + 3)(x + 5)
  40. 5929

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