← Identities, polynomials and quadratic roots lesson New set →

Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 78 · 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 = 14 and ab = 45, what is (a − b)²? (a) 196 (b) 4 (c) 106 (d) 16
  2. 2.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4634 (b) 4624 (c) 4896 (d) 5184
  3. 3.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3364 (b) 3854 (c) 3844 (d) 3604
  4. 4.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10404 (b) 10414 (c) 9604 (d) 10004
  5. 5.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 181 (c) 541 (d) 361
  6. 6.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x − 8)(x − 9) (c) (x + 8)(x + 9) (d) (x + 1)(x + 16)
  7. 7.If a + b = 21 and ab = 108, what is (a − b)²? (a) 3 (b) 441 (c) 9 (d) 225
  8. 8.What is the sum of the roots of x² − 11x + 30 = 0? (a) 12 (b) 10 (c) 30 (d) 11
  9. 9.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 901 (b) 841 (c) 971 (d) 961
  10. 10.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 7)(x + 9) (c) (x + 1)(x + 15) (d) (x − 7)(x − 9)
  11. 11.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 289 (c) 217 (d) 145
  12. 12.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x − 6)(x − 7) (c) (x + 6)(x + 7) (d) (x + 1)(x + 12)
  13. 13.What is the product of the roots of x² − 6x + 8 = 0? (a) 7 (b) 6 (c) 9 (d) 8
  14. 14.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1166 (b) 1156 (c) 676 (d) 916
  15. 15.What is the product of the roots of x² − 8x + 12 = 0? (a) 11 (b) 8 (c) 13 (d) 12
  16. 16.Which is the correct factorisation of x² + 13x + 36? (a) (x + 4)(x + 9) (b) (x − 4)(x − 9) (c) (x + 1)(x + 36) (d) (x + 1)(x + 12)
  17. 17.If a + b = 10 and ab = 16, what is a² + b²? (a) 100 (b) 132 (c) 68 (d) 84
  18. 18.What is the sum of the roots of x² − 5x + 6 = 0? (a) 6 (b) 5 (c) 7 (d) 4
  19. 19.What are the roots of x² − 12x + 32 = 0? (a) 1 and 32 (b) 2 and 16 (c) 1 and 11 (d) 4 and 8
  20. 20.What is the product of the roots of x² − 8x + 15 = 0? (a) 16 (b) 15 (c) 14 (d) 8
  21. 21.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x − 2)(x − 6) (c) (x + 2)(x + 6) (d) (x + 1)(x + 12)
  22. 22.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x + 1)(x + 27) (c) (x + 3)(x + 9) (d) (x − 3)(x − 9)
  23. 23.If a + b = 16 and ab = 60, what is a² + b²? (a) 196 (b) 136 (c) 376 (d) 256
  24. 24.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 1)(x + 5) (c) (x − 2)(x − 4) (d) (x + 2)(x + 4)
  25. 25.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7235 (b) 7225 (c) 5625 (d) 6425
  26. 26.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 8 and 9 (c) 1 and 16 (d) 2 and 36
  27. 27.If a + b = 18 and ab = 72, what is a² + b²? (a) 324 (b) 252 (c) 468 (d) 180
  28. 28.What are the roots of x² − 11x + 30 = 0? (a) 1 and 30 (b) 2 and 15 (c) 5 and 6 (d) 1 and 10
  29. 29.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6084 (b) 6724 (c) 6396 (d) 6094
  30. 30.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)
  31. 31.What is the product of the roots of x² − 10x + 16 = 0? (a) 17 (b) 10 (c) 16 (d) 15
  32. 32.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 1)(x + 8) (c) (x + 1)(x + 14) (d) (x + 2)(x + 7)
  33. 33.If a + b = 9 and ab = 20, what is (a − b)²? (a) 41 (b) 61 (c) 81 (d) 1
  34. 34.What is the product of the roots of x² − 12x + 27 = 0? (a) 12 (b) 27 (c) 26 (d) 28
  35. 35.If a + b = 18 and ab = 80, what is a² + b²? (a) 244 (b) 324 (c) 164 (d) 484
  36. 36.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8846 (b) 7396 (c) 8116 (d) 8836
  37. 37.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8464 (b) 8104 (c) 8474 (d) 7744
  38. 38.What are the roots of x² − 10x + 16 = 0? (a) 1 and 16 (b) 2 and 8 (c) 1 and 9 (d) 3 and 7
  39. 39.What is the sum of the roots of x² − 7x + 10 = 0? (a) 7 (b) 10 (c) 6 (d) 8
  40. 40.If a + b = 19 and ab = 84, what is a² + b²? (a) 277 (b) 529 (c) 361 (d) 193

Answer key

  1. 16
  2. 4624
  3. 3844
  4. 10404
  5. 181
  6. (x + 8)(x + 9)
  7. 9
  8. 11
  9. 961
  10. (x + 7)(x + 9)
  11. 145
  12. (x + 6)(x + 7)
  13. 8
  14. 1156
  15. 12
  16. (x + 4)(x + 9)
  17. 68
  18. 5
  19. 4 and 8
  20. 15
  21. (x + 2)(x + 6)
  22. (x + 3)(x + 9)
  23. 136
  24. (x + 2)(x + 4)
  25. 7225
  26. 8 and 9
  27. 180
  28. 5 and 6
  29. 6084
  30. (x + 7)(x + 8)
  31. 16
  32. (x + 2)(x + 7)
  33. 1
  34. 27
  35. 164
  36. 8836
  37. 8464
  38. 2 and 8
  39. 7
  40. 193

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