← Identities, polynomials and quadratic roots lesson New set →

Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 412 · 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 = 7 and ab = 6, what is (a − b)²? (a) 37 (b) 25 (c) 49 (d) 5
  2. 2.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 63) (c) (x + 7)(x + 9) (d) (x + 1)(x + 15)
  3. 3.If a + b = 15 and ab = 56, what is (a − b)²? (a) 1 (b) 169 (c) 113 (d) 225
  4. 4.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1601 (b) 1521 (c) 1691 (d) 1681
  5. 5.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4899 (b) 4761 (c) 4771 (d) 5041
  6. 6.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x + 1)(x + 35) (c) (x − 5)(x − 7) (d) (x + 5)(x + 7)
  7. 7.What is the sum of the roots of x² − 11x + 30 = 0? (a) 30 (b) 12 (c) 10 (d) 11
  8. 8.If a + b = 19 and ab = 90, what is a² + b²? (a) 181 (b) 271 (c) 361 (d) 541
  9. 9.If a + b = 11 and ab = 30, what is (a − b)²? (a) 1 (b) 121 (c) 91 (d) 61
  10. 10.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2611 (b) 2501 (c) 2401 (d) 2601
  11. 11.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 56 (c) 14 (d) 15
  12. 12.If a + b = 10 and ab = 24, what is (a − b)²? (a) 2 (b) 52 (c) 4 (d) 100
  13. 13.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 17 (c) 16 (d) 63
  14. 14.What is the product of the roots of x² − 8x + 15 = 0? (a) 16 (b) 8 (c) 15 (d) 14
  15. 15.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 17 (c) 72 (d) 71
  16. 16.If a + b = 21 and ab = 108, what is (a − b)²? (a) 9 (b) 441 (c) 3 (d) 225
  17. 17.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10004 (b) 10414 (c) 9604 (d) 10404
  18. 18.If a + b = 11 and ab = 24, what is a² + b²? (a) 169 (b) 73 (c) 121 (d) 97
  19. 19.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 3035 (b) 2025 (c) 3025 (d) 2525
  20. 20.What is the sum of the roots of x² − 10x + 16 = 0? (a) 10 (b) 9 (c) 16 (d) 11
  21. 21.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x + 1)(x + 42) (c) (x − 6)(x − 7) (d) (x + 1)(x + 12)
  22. 22.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 404 (b) 324 (c) 494 (d) 484
  23. 23.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6251 (b) 6241 (c) 6561 (d) 6399
  24. 24.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x − 5)(x − 9) (c) (x + 5)(x + 9) (d) (x + 1)(x + 45)
  25. 25.If a + b = 14 and ab = 48, what is (a − b)²? (a) 196 (b) 100 (c) 2 (d) 4
  26. 26.If a + b = 21 and ab = 108, what is a² + b²? (a) 657 (b) 333 (c) 225 (d) 441
  27. 27.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x + 1)(x + 13) (c) (x + 6)(x + 8) (d) (x − 6)(x − 8)
  28. 28.If a + b = 9 and ab = 18, what is (a − b)²? (a) 3 (b) 9 (c) 45 (d) 81
  29. 29.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x + 1)(x + 8) (c) (x − 4)(x − 5) (d) (x + 4)(x + 5)
  30. 30.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x + 3)(x + 5) (c) (x − 3)(x − 5) (d) (x + 1)(x + 15)
  31. 31.What is the product of the roots of x² − 8x + 12 = 0? (a) 13 (b) 8 (c) 11 (d) 12
  32. 32.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 16)
  33. 33.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 896 (b) 794 (c) 1024 (d) 784
  34. 34.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 4)(x + 7) (c) (x + 1)(x + 10) (d) (x + 1)(x + 28)
  35. 35.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1024 (b) 784 (c) 904 (d) 1034
  36. 36.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x − 4)(x − 6) (c) (x + 4)(x + 6) (d) (x + 1)(x + 9)
  37. 37.If a + b = 5 and ab = 4, what is a² + b²? (a) 33 (b) 21 (c) 17 (d) 25
  38. 38.What is the sum of the roots of x² − 12x + 32 = 0? (a) 11 (b) 12 (c) 13 (d) 32
  39. 39.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 16 (c) 18 (d) 17
  40. 40.If a + b = 16 and ab = 55, what is a² + b²? (a) 146 (b) 366 (c) 201 (d) 256

Answer key

  1. 25
  2. (x + 7)(x + 9)
  3. 1
  4. 1681
  5. 4761
  6. (x + 5)(x + 7)
  7. 11
  8. 181
  9. 1
  10. 2601
  11. 15
  12. 4
  13. 16
  14. 15
  15. 72
  16. 9
  17. 10404
  18. 73
  19. 3025
  20. 10
  21. (x + 6)(x + 7)
  22. 484
  23. 6241
  24. (x + 5)(x + 9)
  25. 4
  26. 225
  27. (x + 6)(x + 8)
  28. 9
  29. (x + 4)(x + 5)
  30. (x + 3)(x + 5)
  31. 12
  32. (x + 8)(x + 9)
  33. 784
  34. (x + 4)(x + 7)
  35. 1024
  36. (x + 4)(x + 6)
  37. 17
  38. 12
  39. 17
  40. 146

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