← Identities, polynomials and quadratic roots lesson New set →

Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 168 · 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 38²? (a) 1444 (b) 1454 (c) 1764 (d) 1596
  2. 2.What is the product of the roots of x² − 13x + 40 = 0? (a) 41 (b) 39 (c) 13 (d) 40
  3. 3.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x − 7)(x − 8) (c) (x + 7)(x + 8) (d) (x + 1)(x + 56)
  4. 4.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 56 (c) 14 (d) 15
  5. 5.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 899 (b) 841 (c) 961 (d) 851
  6. 6.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x − 5)(x − 8) (c) (x + 1)(x + 40) (d) (x + 5)(x + 8)
  7. 7.What is the sum of the roots of x² − 12x + 27 = 0? (a) 27 (b) 11 (c) 12 (d) 13
  8. 8.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2509 (b) 2809 (c) 2819 (d) 2209
  9. 9.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x − 6)(x − 8) (c) (x + 6)(x + 8) (d) (x + 1)(x + 48)
  10. 10.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x + 1)(x + 11) (c) (x − 5)(x − 7) (d) (x + 5)(x + 7)
  11. 11.If a + b = 6 and ab = 8, what is (a − b)²? (a) 20 (b) 36 (c) 2 (d) 4
  12. 12.If a + b = 20 and ab = 96, what is a² + b²? (a) 304 (b) 208 (c) 400 (d) 592
  13. 13.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
  14. 14.What are the roots of x² − 9x + 18 = 0? (a) 2 and 9 (b) 3 and 6 (c) 1 and 8 (d) 1 and 18
  15. 15.What is the sum of the roots of x² − 6x + 8 = 0? (a) 7 (b) 8 (c) 6 (d) 5
  16. 16.If a + b = 15 and ab = 44, what is a² + b²? (a) 225 (b) 313 (c) 137 (d) 181
  17. 17.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x + 1)(x + 36) (c) (x − 4)(x − 9) (d) (x + 4)(x + 9)
  18. 18.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 1024 (c) 896 (d) 784
  19. 19.If a + b = 17 and ab = 60, what is (a − b)²? (a) 7 (b) 49 (c) 169 (d) 289
  20. 20.If a + b = 9 and ab = 18, what is (a − b)²? (a) 81 (b) 45 (c) 3 (d) 9
  21. 21.If a + b = 9 and ab = 20, what is a² + b²? (a) 41 (b) 121 (c) 61 (d) 81
  22. 22.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4916 (b) 5486 (c) 4356 (d) 5476
  23. 23.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x − 8)(x − 9)
  24. 24.Which is the correct factorisation of x² + 7x + 10? (a) (x + 2)(x + 5) (b) (x − 2)(x − 5) (c) (x + 1)(x + 6) (d) (x + 1)(x + 10)
  25. 25.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 361 (b) 441 (c) 401 (d) 451
  26. 26.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 10404 (b) 9604 (c) 9996 (d) 9614
  27. 27.What are the roots of x² − 9x + 14 = 0? (a) 3 and 6 (b) 2 and 7 (c) 1 and 8 (d) 1 and 14
  28. 28.What are the roots of x² − 15x + 54 = 0? (a) 1 and 54 (b) 6 and 9 (c) 1 and 14 (d) 2 and 27
  29. 29.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 399 (b) 441 (c) 371 (d) 361
  30. 30.If a + b = 14 and ab = 33, what is a² + b²? (a) 130 (b) 262 (c) 196 (d) 163
  31. 31.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 739 (b) 891 (c) 1089 (d) 729
  32. 32.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 841 (b) 971 (c) 961 (d) 901
  33. 33.What is the sum of the roots of x² − 13x + 40 = 0? (a) 14 (b) 12 (c) 40 (d) 13
  34. 34.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3616 (b) 4096 (c) 4106 (d) 3136
  35. 35.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 73 (c) 17 (d) 71
  36. 36.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 217 (c) 289 (d) 145
  37. 37.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x − 4)(x − 7) (c) (x + 4)(x + 7) (d) (x + 1)(x + 28)
  38. 38.If a + b = 11 and ab = 10, what is (a − b)²? (a) 101 (b) 9 (c) 81 (d) 121
  39. 39.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1444 (b) 1774 (c) 1764 (d) 1604
  40. 40.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3146 (b) 3584 (c) 4096 (d) 3136

Answer key

  1. 1444
  2. 40
  3. (x + 7)(x + 8)
  4. 15
  5. 841
  6. (x + 5)(x + 8)
  7. 12
  8. 2809
  9. (x + 6)(x + 8)
  10. (x + 5)(x + 7)
  11. 4
  12. 208
  13. 7 and 8
  14. 3 and 6
  15. 6
  16. 137
  17. (x + 4)(x + 9)
  18. 784
  19. 49
  20. 9
  21. 41
  22. 5476
  23. (x + 8)(x + 9)
  24. (x + 2)(x + 5)
  25. 441
  26. 9604
  27. 2 and 7
  28. 6 and 9
  29. 361
  30. 130
  31. 729
  32. 961
  33. 13
  34. 4096
  35. 72
  36. 145
  37. (x + 4)(x + 7)
  38. 81
  39. 1764
  40. 3136

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