← Identities, polynomials and quadratic roots lesson New set →

Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 5 · 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.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 3035 (b) 2525 (c) 2025 (d) 3025
  2. 2.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1225 (b) 2025 (c) 1625 (d) 2035
  3. 3.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 18 (d) 16
  4. 4.What is the product of the roots of x² − 15x + 54 = 0? (a) 53 (b) 54 (c) 55 (d) 15
  5. 5.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 63) (c) (x + 1)(x + 15) (d) (x + 7)(x + 9)
  6. 6.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 484 (b) 324 (c) 334 (d) 396
  7. 7.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x − 7)(x − 8) (c) (x + 7)(x + 8) (d) (x + 1)(x + 14)
  8. 8.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2704 (b) 2304 (c) 2504 (d) 2714
  9. 9.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x + 1)(x + 12) (c) (x + 5)(x + 8) (d) (x − 5)(x − 8)
  10. 10.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 1 (c) 181 (d) 361
  11. 11.If a + b = 11 and ab = 28, what is a² + b²? (a) 121 (b) 93 (c) 177 (d) 65
  12. 12.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 63 (c) 16 (d) 15
  13. 13.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3374 (b) 3596 (c) 3364 (d) 3844
  14. 14.If a + b = 15 and ab = 50, what is a² + b²? (a) 125 (b) 175 (c) 325 (d) 225
  15. 15.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 1)(x + 20) (c) (x + 1)(x + 8) (d) (x + 4)(x + 5)
  16. 16.What is the sum of the roots of x² − 11x + 30 = 0? (a) 11 (b) 12 (c) 30 (d) 10
  17. 17.What is the product of the roots of x² − 6x + 8 = 0? (a) 6 (b) 9 (c) 8 (d) 7
  18. 18.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10211 (b) 9801 (c) 10001 (d) 10201
  19. 19.What is the product of the roots of x² − 10x + 24 = 0? (a) 24 (b) 10 (c) 25 (d) 23
  20. 20.What are the roots of x² − 11x + 28 = 0? (a) 2 and 14 (b) 4 and 7 (c) 1 and 28 (d) 1 and 10
  21. 21.What is the product of the roots of x² − 9x + 20 = 0? (a) 9 (b) 20 (c) 21 (d) 19
  22. 22.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 739 (b) 891 (c) 1089 (d) 729
  23. 23.If a + b = 11 and ab = 28, what is (a − b)²? (a) 121 (b) 3 (c) 9 (d) 65
  24. 24.If a + b = 5 and ab = 6, what is a² + b²? (a) 13 (b) 19 (c) 37 (d) 25
  25. 25.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1601 (b) 1521 (c) 1681 (d) 1691
  26. 26.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 1)(x + 72)
  27. 27.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 9025 (b) 7235 (c) 8075 (d) 7225
  28. 28.If a + b = 15 and ab = 56, what is a² + b²? (a) 337 (b) 169 (c) 113 (d) 225
  29. 29.What is the sum of the roots of x² − 14x + 45 = 0? (a) 14 (b) 13 (c) 45 (d) 15
  30. 30.What is the sum of the roots of x² − 14x + 48 = 0? (a) 15 (b) 14 (c) 48 (d) 13
  31. 31.If a + b = 18 and ab = 72, what is a² + b²? (a) 180 (b) 324 (c) 468 (d) 252
  32. 32.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 9604 (b) 10004 (c) 10414 (d) 10404
  33. 33.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 15 (c) 57 (d) 56
  34. 34.If a + b = 18 and ab = 80, what is a² + b²? (a) 324 (b) 244 (c) 164 (d) 484
  35. 35.What is the product of the roots of x² − 13x + 40 = 0? (a) 41 (b) 13 (c) 40 (d) 39
  36. 36.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4624 (b) 5194 (c) 4904 (d) 5184
  37. 37.If a + b = 6 and ab = 5, what is a² + b²? (a) 31 (b) 46 (c) 36 (d) 26
  38. 38.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2819 (b) 2509 (c) 2209 (d) 2809
  39. 39.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3259 (b) 3591 (c) 3249 (d) 3969
  40. 40.What is the product of the roots of x² − 16x + 63 = 0? (a) 16 (b) 63 (c) 64 (d) 62

Answer key

  1. 3025
  2. 2025
  3. 17
  4. 54
  5. (x + 7)(x + 9)
  6. 324
  7. (x + 7)(x + 8)
  8. 2704
  9. (x + 5)(x + 8)
  10. 1
  11. 65
  12. 16
  13. 3364
  14. 125
  15. (x + 4)(x + 5)
  16. 11
  17. 8
  18. 10201
  19. 24
  20. 4 and 7
  21. 20
  22. 729
  23. 9
  24. 13
  25. 1681
  26. (x + 8)(x + 9)
  27. 7225
  28. 113
  29. 14
  30. 14
  31. 180
  32. 10404
  33. 56
  34. 164
  35. 40
  36. 5184
  37. 26
  38. 2809
  39. 3249
  40. 63

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