← Identities, polynomials and quadratic roots lesson New set →

Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 836 · 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 = 11 and ab = 30, what is (a − b)²? (a) 91 (b) 121 (c) 61 (d) 1
  2. 2.If a + b = 9 and ab = 18, what is a² + b²? (a) 117 (b) 45 (c) 63 (d) 81
  3. 3.What are the roots of x² − 16x + 63 = 0? (a) 3 and 21 (b) 7 and 9 (c) 1 and 15 (d) 1 and 63
  4. 4.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 63 (c) 62 (d) 16
  5. 5.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x + 1)(x + 6) (c) (x + 2)(x + 5) (d) (x − 2)(x − 5)
  6. 6.If a + b = 17 and ab = 66, what is (a − b)²? (a) 157 (b) 5 (c) 289 (d) 25
  7. 7.If a + b = 5 and ab = 4, what is a² + b²? (a) 33 (b) 25 (c) 21 (d) 17
  8. 8.What is the product of the roots of x² − 13x + 42 = 0? (a) 13 (b) 42 (c) 43 (d) 41
  9. 9.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)
  10. 10.If a + b = 3 and ab = 2, what is a² + b²? (a) 9 (b) 7 (c) 13 (d) 5
  11. 11.What is the sum of the roots of x² − 6x + 8 = 0? (a) 5 (b) 7 (c) 6 (d) 8
  12. 12.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1936 (b) 1296 (c) 1946 (d) 1616
  13. 13.What are the roots of x² − 6x + 8 = 0? (a) 2 and 4 (b) 1 and 5 (c) 2 and 5 (d) 1 and 8
  14. 14.If a + b = 12 and ab = 35, what is (a − b)²? (a) 74 (b) 144 (c) 4 (d) 2
  15. 15.What is the sum of the roots of x² − 13x + 40 = 0? (a) 40 (b) 13 (c) 14 (d) 12
  16. 16.If a + b = 19 and ab = 90, what is a² + b²? (a) 181 (b) 541 (c) 361 (d) 271
  17. 17.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 5776 (b) 6416 (c) 7056 (d) 7066
  18. 18.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 375 (b) 225 (c) 625 (d) 235
  19. 19.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1099 (b) 1089 (c) 909 (d) 729
  20. 20.If a + b = 15 and ab = 50, what is (a − b)²? (a) 25 (b) 5 (c) 225 (d) 125
  21. 21.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 904 (b) 1024 (c) 1034 (d) 784
  22. 22.If a + b = 10 and ab = 21, what is (a − b)²? (a) 16 (b) 4 (c) 100 (d) 58
  23. 23.If a + b = 18 and ab = 80, what is a² + b²? (a) 244 (b) 164 (c) 324 (d) 484
  24. 24.If a + b = 11 and ab = 10, what is (a − b)²? (a) 101 (b) 9 (c) 121 (d) 81
  25. 25.If a + b = 6 and ab = 5, what is a² + b²? (a) 36 (b) 26 (c) 31 (d) 46
  26. 26.What are the roots of x² − 10x + 21 = 0? (a) 2 and 8 (b) 3 and 7 (c) 1 and 21 (d) 1 and 9
  27. 27.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x − 4)(x − 8) (c) (x + 4)(x + 8) (d) (x + 1)(x + 11)
  28. 28.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 9025 (b) 8075 (c) 7235 (d) 7225
  29. 29.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 6)(x + 7) (c) (x − 6)(x − 7) (d) (x + 1)(x + 42)
  30. 30.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3854 (b) 3604 (c) 3364 (d) 3844
  31. 31.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 5)(x + 6) (c) (x − 5)(x − 6) (d) (x + 1)(x + 30)
  32. 32.What is the sum of the roots of x² − 13x + 36 = 0? (a) 12 (b) 14 (c) 13 (d) 36
  33. 33.What is the sum of the roots of x² − 11x + 24 = 0? (a) 24 (b) 10 (c) 12 (d) 11
  34. 34.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2501 (b) 2611 (c) 2601 (d) 2401
  35. 35.Which is the correct factorisation of x² + 8x + 12? (a) (x + 2)(x + 6) (b) (x − 2)(x − 6) (c) (x + 1)(x + 7) (d) (x + 1)(x + 12)
  36. 36.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9035 (b) 11025 (c) 9025 (d) 9975
  37. 37.If a + b = 17 and ab = 70, what is (a − b)²? (a) 289 (b) 149 (c) 9 (d) 3
  38. 38.What is the product of the roots of x² − 10x + 21 = 0? (a) 22 (b) 20 (c) 21 (d) 10
  39. 39.If a + b = 14 and ab = 40, what is (a − b)²? (a) 6 (b) 196 (c) 36 (d) 116
  40. 40.If a + b = 8 and ab = 15, what is a² + b²? (a) 34 (b) 49 (c) 64 (d) 94

Answer key

  1. 1
  2. 45
  3. 7 and 9
  4. 63
  5. (x + 2)(x + 5)
  6. 25
  7. 17
  8. 42
  9. (x + 8)(x + 9)
  10. 5
  11. 6
  12. 1936
  13. 2 and 4
  14. 4
  15. 13
  16. 181
  17. 7056
  18. 225
  19. 1089
  20. 25
  21. 1024
  22. 16
  23. 164
  24. 81
  25. 26
  26. 3 and 7
  27. (x + 4)(x + 8)
  28. 7225
  29. (x + 6)(x + 7)
  30. 3844
  31. (x + 5)(x + 6)
  32. 13
  33. 11
  34. 2601
  35. (x + 2)(x + 6)
  36. 9025
  37. 9
  38. 21
  39. 36
  40. 34

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