← Identities, polynomials and quadratic roots lesson New set →

Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 634 · 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 = 17 and ab = 70, what is a² + b²? (a) 429 (b) 289 (c) 219 (d) 149
  2. 2.If a + b = 8 and ab = 15, what is a² + b²? (a) 49 (b) 94 (c) 34 (d) 64
  3. 3.If a + b = 12 and ab = 27, what is a² + b²? (a) 117 (b) 90 (c) 144 (d) 198
  4. 4.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 72 (c) 73 (d) 17
  5. 5.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 9604 (b) 10004 (c) 10414 (d) 10404
  6. 6.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x − 5)(x − 7) (c) (x + 5)(x + 7) (d) (x + 1)(x + 11)
  7. 7.What is the product of the roots of x² − 5x + 6 = 0? (a) 7 (b) 8 (c) 6 (d) 5
  8. 8.What is the product of the roots of x² − 13x + 42 = 0? (a) 42 (b) 13 (c) 41 (d) 43
  9. 9.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 1)(x + 72) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
  10. 10.If a + b = 8 and ab = 15, what is (a − b)²? (a) 2 (b) 64 (c) 4 (d) 34
  11. 11.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 202 (c) 301 (d) 400
  12. 12.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 6375 (b) 7225 (c) 5635 (d) 5625
  13. 13.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3616 (b) 4106 (c) 3136 (d) 4096
  14. 14.What is the product of the roots of x² − 15x + 56 = 0? (a) 56 (b) 15 (c) 57 (d) 55
  15. 15.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 1)(x + 56) (c) (x + 7)(x + 8) (d) (x − 7)(x − 8)
  16. 16.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 529 (b) 391 (c) 299 (d) 289
  17. 17.What is the sum of the roots of x² − 10x + 24 = 0? (a) 9 (b) 24 (c) 11 (d) 10
  18. 18.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 15 (c) 14 (d) 13
  19. 19.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 7396 (b) 8846 (c) 8836 (d) 8116
  20. 20.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x − 2)(x − 9) (c) (x + 2)(x + 9) (d) (x + 1)(x + 18)
  21. 21.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 961 (b) 841 (c) 901 (d) 971
  22. 22.If a + b = 15 and ab = 50, what is (a − b)²? (a) 25 (b) 5 (c) 125 (d) 225
  23. 23.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4366 (b) 4884 (c) 5476 (d) 4356
  24. 24.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 1 and 16 (c) 2 and 36 (d) 8 and 9
  25. 25.If a + b = 16 and ab = 63, what is (a − b)²? (a) 2 (b) 256 (c) 4 (d) 130
  26. 26.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 7579 (b) 8649 (c) 8091 (d) 7569
  27. 27.If a + b = 16 and ab = 60, what is a² + b²? (a) 376 (b) 196 (c) 256 (d) 136
  28. 28.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 1)(x + 9) (c) (x + 4)(x + 6) (d) (x + 1)(x + 24)
  29. 29.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2304 (b) 2314 (c) 2496 (d) 2704
  30. 30.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 676 (b) 884 (c) 686 (d) 1156
  31. 31.If a + b = 7 and ab = 6, what is (a − b)²? (a) 49 (b) 5 (c) 25 (d) 37
  32. 32.If a + b = 17 and ab = 70, what is (a − b)²? (a) 149 (b) 289 (c) 3 (d) 9
  33. 33.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 4)(x + 7) (c) (x + 1)(x + 28) (d) (x + 1)(x + 10)
  34. 34.What are the roots of x² − 13x + 42 = 0? (a) 1 and 42 (b) 6 and 7 (c) 2 and 21 (d) 1 and 12
  35. 35.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 10025 (b) 11035 (c) 11025 (d) 9025
  36. 36.If a + b = 17 and ab = 66, what is (a − b)²? (a) 25 (b) 289 (c) 5 (d) 157
  37. 37.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x − 6)(x − 8) (c) (x + 6)(x + 8) (d) (x + 1)(x + 13)
  38. 38.If a + b = 11 and ab = 10, what is a² + b²? (a) 101 (b) 141 (c) 121 (d) 111
  39. 39.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5041 (b) 4901 (c) 4761 (d) 5051
  40. 40.If a + b = 13 and ab = 36, what is (a − b)²? (a) 169 (b) 5 (c) 25 (d) 97

Answer key

  1. 149
  2. 34
  3. 90
  4. 72
  5. 10404
  6. (x + 5)(x + 7)
  7. 6
  8. 42
  9. (x + 8)(x + 9)
  10. 4
  11. 202
  12. 5625
  13. 4096
  14. 56
  15. (x + 7)(x + 8)
  16. 289
  17. 10
  18. 14
  19. 8836
  20. (x + 2)(x + 9)
  21. 961
  22. 25
  23. 4356
  24. 8 and 9
  25. 4
  26. 7569
  27. 136
  28. (x + 4)(x + 6)
  29. 2304
  30. 676
  31. 25
  32. 9
  33. (x + 4)(x + 7)
  34. 6 and 7
  35. 11025
  36. 25
  37. (x + 6)(x + 8)
  38. 101
  39. 5041
  40. 25

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