← Identities, polynomials and quadratic roots lesson New set →

Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 442 · 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 = 4 and ab = 3, what is (a − b)²? (a) 10 (b) 16 (c) 2 (d) 4
  2. 2.If a + b = 7 and ab = 12, what is (a − b)²? (a) 1 (b) 37 (c) 25 (d) 49
  3. 3.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6889 (b) 6391 (c) 5939 (d) 5929
  4. 4.What is the sum of the roots of x² − 13x + 36 = 0? (a) 14 (b) 12 (c) 13 (d) 36
  5. 5.What is the sum of the roots of x² − 5x + 6 = 0? (a) 5 (b) 7 (c) 6 (d) 4
  6. 6.What is the sum of the roots of x² − 10x + 24 = 0? (a) 24 (b) 9 (c) 11 (d) 10
  7. 7.If a + b = 5 and ab = 6, what is (a − b)²? (a) 25 (b) 1 (c) 13 (d) 19
  8. 8.If a + b = 18 and ab = 80, what is a² + b²? (a) 484 (b) 244 (c) 324 (d) 164
  9. 9.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6561 (b) 6241 (c) 6251 (d) 6399
  10. 10.If a + b = 10 and ab = 16, what is (a − b)²? (a) 100 (b) 68 (c) 36 (d) 6
  11. 11.What is the product of the roots of x² − 13x + 36 = 0? (a) 13 (b) 37 (c) 35 (d) 36
  12. 12.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 16 (c) 56 (d) 15
  13. 13.If a + b = 12 and ab = 20, what is a² + b²? (a) 124 (b) 184 (c) 144 (d) 104
  14. 14.Which is the correct factorisation of x² + 11x + 18? (a) (x − 2)(x − 9) (b) (x + 2)(x + 9) (c) (x + 1)(x + 10) (d) (x + 1)(x + 18)
  15. 15.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2499 (b) 2601 (c) 2411 (d) 2401
  16. 16.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 56) (c) (x + 7)(x + 8) (d) (x + 1)(x + 14)
  17. 17.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 6416 (b) 7056 (c) 7066 (d) 5776
  18. 18.If a + b = 16 and ab = 48, what is a² + b²? (a) 352 (b) 208 (c) 256 (d) 160
  19. 19.If a + b = 16 and ab = 60, what is a² + b²? (a) 376 (b) 196 (c) 256 (d) 136
  20. 20.If a + b = 12 and ab = 35, what is (a − b)²? (a) 4 (b) 144 (c) 2 (d) 74
  21. 21.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 1)(x + 20) (c) (x + 4)(x + 5) (d) (x + 1)(x + 8)
  22. 22.If a + b = 13 and ab = 36, what is a² + b²? (a) 133 (b) 97 (c) 169 (d) 241
  23. 23.Which is the correct factorisation of x² + 12x + 27? (a) (x + 3)(x + 9) (b) (x + 1)(x + 11) (c) (x + 1)(x + 27) (d) (x − 3)(x − 9)
  24. 24.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6084 (b) 6404 (c) 6734 (d) 6724
  25. 25.If a + b = 9 and ab = 14, what is a² + b²? (a) 81 (b) 109 (c) 67 (d) 53
  26. 26.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x − 5)(x − 7) (c) (x + 1)(x + 35) (d) (x + 5)(x + 7)
  27. 27.If a + b = 20 and ab = 99, what is (a − b)²? (a) 202 (b) 2 (c) 400 (d) 4
  28. 28.What is the product of the roots of x² − 12x + 27 = 0? (a) 12 (b) 27 (c) 26 (d) 28
  29. 29.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2401 (b) 2611 (c) 2501 (d) 2601
  30. 30.If a + b = 7 and ab = 10, what is (a − b)²? (a) 3 (b) 29 (c) 9 (d) 49
  31. 31.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)
  32. 32.If a + b = 13 and ab = 40, what is a² + b²? (a) 129 (b) 89 (c) 169 (d) 249
  33. 33.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 441 (b) 399 (c) 371 (d) 361
  34. 34.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 4096 (b) 3584 (c) 3146 (d) 3136
  35. 35.If a + b = 9 and ab = 8, what is a² + b²? (a) 81 (b) 73 (c) 65 (d) 97
  36. 36.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 1)(x + 5) (c) (x + 2)(x + 4) (d) (x + 1)(x + 8)
  37. 37.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x − 6)(x − 9) (c) (x + 6)(x + 9) (d) (x + 1)(x + 54)
  38. 38.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 72 (c) 17 (d) 71
  39. 39.If a + b = 15 and ab = 50, what is a² + b²? (a) 225 (b) 325 (c) 175 (d) 125
  40. 40.If a + b = 20 and ab = 96, what is a² + b²? (a) 400 (b) 592 (c) 304 (d) 208

Answer key

  1. 4
  2. 1
  3. 5929
  4. 13
  5. 5
  6. 10
  7. 1
  8. 164
  9. 6241
  10. 36
  11. 36
  12. 15
  13. 104
  14. (x + 2)(x + 9)
  15. 2401
  16. (x + 7)(x + 8)
  17. 7056
  18. 160
  19. 136
  20. 4
  21. (x + 4)(x + 5)
  22. 97
  23. (x + 3)(x + 9)
  24. 6724
  25. 53
  26. (x + 5)(x + 7)
  27. 4
  28. 27
  29. 2601
  30. 9
  31. (x + 7)(x + 9)
  32. 89
  33. 361
  34. 3136
  35. 65
  36. (x + 2)(x + 4)
  37. (x + 6)(x + 9)
  38. 72
  39. 125
  40. 208

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