← Identities, polynomials and quadratic roots lesson New set →

Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 205 · 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 − 17x + 72 = 0? (a) 16 (b) 72 (c) 17 (d) 18
  2. 2.What is the sum of the roots of x² − 7x + 12 = 0? (a) 6 (b) 12 (c) 8 (d) 7
  3. 3.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 851 (b) 961 (c) 841 (d) 899
  4. 4.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 451 (b) 401 (c) 441 (d) 361
  5. 5.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 625 (b) 925 (c) 1235 (d) 1225
  6. 6.If a + b = 20 and ab = 96, what is (a − b)²? (a) 4 (b) 400 (c) 16 (d) 208
  7. 7.What are the roots of x² − 15x + 56 = 0? (a) 2 and 28 (b) 1 and 14 (c) 7 and 8 (d) 1 and 56
  8. 8.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2611 (b) 2401 (c) 2501 (d) 2601
  9. 9.What is the product of the roots of x² − 12x + 27 = 0? (a) 26 (b) 28 (c) 27 (d) 12
  10. 10.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 40) (c) (x + 5)(x + 8) (d) (x + 1)(x + 12)
  11. 11.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 361 (b) 399 (c) 371 (d) 441
  12. 12.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2304 (b) 2504 (c) 2704 (d) 2714
  13. 13.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 739 (b) 1089 (c) 729 (d) 891
  14. 14.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 17 (c) 72 (d) 71
  15. 15.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6409 (b) 5929 (c) 6899 (d) 6889
  16. 16.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3731 (b) 3601 (c) 3481 (d) 3721
  17. 17.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 9801 (b) 10001 (c) 10211 (d) 10201
  18. 18.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1521 (b) 1691 (c) 1601 (d) 1681
  19. 19.If a + b = 19 and ab = 88, what is a² + b²? (a) 361 (b) 537 (c) 273 (d) 185
  20. 20.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 4)(x + 6) (c) (x + 1)(x + 24) (d) (x + 1)(x + 9)
  21. 21.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2209 (b) 2819 (c) 2809 (d) 2509
  22. 22.What are the roots of x² − 5x + 6 = 0? (a) 2 and 4 (b) 2 and 3 (c) 1 and 6 (d) 1 and 4
  23. 23.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 2)(x + 3) (c) (x + 1)(x + 4) (d) (x + 1)(x + 6)
  24. 24.What is the product of the roots of x² − 14x + 48 = 0? (a) 49 (b) 14 (c) 47 (d) 48
  25. 25.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 11) (b) (x − 4)(x − 8) (c) (x + 1)(x + 32) (d) (x + 4)(x + 8)
  26. 26.If a + b = 16 and ab = 63, what is a² + b²? (a) 382 (b) 193 (c) 256 (d) 130
  27. 27.What are the roots of x² − 7x + 10 = 0? (a) 1 and 6 (b) 1 and 10 (c) 2 and 5 (d) 3 and 4
  28. 28.If a + b = 14 and ab = 40, what is a² + b²? (a) 156 (b) 276 (c) 116 (d) 196
  29. 29.What is the sum of the roots of x² − 12x + 27 = 0? (a) 27 (b) 11 (c) 12 (d) 13
  30. 30.If a + b = 15 and ab = 50, what is a² + b²? (a) 175 (b) 225 (c) 125 (d) 325
  31. 31.If a + b = 15 and ab = 56, what is a² + b²? (a) 337 (b) 113 (c) 169 (d) 225
  32. 32.If a + b = 3 and ab = 2, what is (a − b)²? (a) 1 (b) 5 (c) 7 (d) 9
  33. 33.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 375 (b) 225 (c) 625 (d) 235
  34. 34.If a + b = 15 and ab = 36, what is a² + b²? (a) 189 (b) 225 (c) 297 (d) 153
  35. 35.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 1)(x + 12) (c) (x − 6)(x − 7) (d) (x + 6)(x + 7)
  36. 36.What is the product of the roots of x² − 11x + 18 = 0? (a) 11 (b) 18 (c) 17 (d) 19
  37. 37.If a + b = 14 and ab = 33, what is a² + b²? (a) 163 (b) 196 (c) 130 (d) 262
  38. 38.If a + b = 13 and ab = 42, what is (a − b)²? (a) 169 (b) 127 (c) 1 (d) 85
  39. 39.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 2)(x + 4) (c) (x − 2)(x − 4) (d) (x + 1)(x + 5)
  40. 40.What is the product of the roots of x² − 13x + 36 = 0? (a) 13 (b) 35 (c) 37 (d) 36

Answer key

  1. 17
  2. 7
  3. 841
  4. 441
  5. 1225
  6. 16
  7. 7 and 8
  8. 2601
  9. 27
  10. (x + 5)(x + 8)
  11. 361
  12. 2704
  13. 729
  14. 72
  15. 6889
  16. 3721
  17. 10201
  18. 1681
  19. 185
  20. (x + 4)(x + 6)
  21. 2809
  22. 2 and 3
  23. (x + 2)(x + 3)
  24. 48
  25. (x + 4)(x + 8)
  26. 130
  27. 2 and 5
  28. 116
  29. 12
  30. 125
  31. 113
  32. 1
  33. 225
  34. 153
  35. (x + 6)(x + 7)
  36. 18
  37. 130
  38. 1
  39. (x + 2)(x + 4)
  40. 36

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