← Identities, polynomials and quadratic roots lesson New set →

Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 294 · 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.What is the sum of the roots of x² − 6x + 8 = 0? (a) 5 (b) 7 (c) 8 (d) 6
  2. 2.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 3025 (b) 2035 (c) 2025 (d) 2475
  3. 3.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4899 (b) 5041 (c) 4761 (d) 4771
  4. 4.If a + b = 12 and ab = 35, what is a² + b²? (a) 144 (b) 214 (c) 74 (d) 109
  5. 5.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4356 (b) 4884 (c) 5476 (d) 4366
  6. 6.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1235 (b) 625 (c) 1225 (d) 925
  7. 7.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x + 1)(x + 8) (c) (x − 4)(x − 5) (d) (x + 1)(x + 20)
  8. 8.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x + 4)(x + 8) (c) (x + 1)(x + 11) (d) (x − 4)(x − 8)
  9. 9.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2916 (b) 2116 (c) 2926 (d) 2516
  10. 10.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 71 (c) 72 (d) 73
  11. 11.If a + b = 8 and ab = 15, what is a² + b²? (a) 94 (b) 64 (c) 49 (d) 34
  12. 12.What are the roots of x² − 13x + 42 = 0? (a) 2 and 21 (b) 6 and 7 (c) 1 and 12 (d) 1 and 42
  13. 13.If a + b = 20 and ab = 99, what is (a − b)²? (a) 202 (b) 4 (c) 2 (d) 400
  14. 14.If a + b = 9 and ab = 8, what is (a − b)²? (a) 7 (b) 49 (c) 81 (d) 65
  15. 15.If a + b = 9 and ab = 14, what is (a − b)²? (a) 53 (b) 81 (c) 5 (d) 25
  16. 16.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 657 (c) 225 (d) 441
  17. 17.What is the product of the roots of x² − 12x + 32 = 0? (a) 33 (b) 31 (c) 32 (d) 12
  18. 18.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x + 1)(x + 16) (c) (x − 2)(x − 8) (d) (x + 2)(x + 8)
  19. 19.If a + b = 20 and ab = 96, what is (a − b)²? (a) 4 (b) 16 (c) 400 (d) 208
  20. 20.What is the sum of the roots of x² − 8x + 15 = 0? (a) 15 (b) 8 (c) 9 (d) 7
  21. 21.If a + b = 11 and ab = 30, what is a² + b²? (a) 91 (b) 181 (c) 61 (d) 121
  22. 22.What are the roots of x² − 13x + 36 = 0? (a) 1 and 12 (b) 1 and 36 (c) 2 and 18 (d) 4 and 9
  23. 23.What are the roots of x² − 12x + 27 = 0? (a) 3 and 9 (b) 2 and 10 (c) 1 and 27 (d) 1 and 11
  24. 24.If a + b = 19 and ab = 88, what is a² + b²? (a) 361 (b) 537 (c) 273 (d) 185
  25. 25.If a + b = 16 and ab = 48, what is (a − b)²? (a) 160 (b) 256 (c) 8 (d) 64
  26. 26.Which is the correct factorisation of x² + 9x + 14? (a) (x + 2)(x + 7) (b) (x + 1)(x + 14) (c) (x − 2)(x − 7) (d) (x + 1)(x + 8)
  27. 27.If a + b = 13 and ab = 42, what is a² + b²? (a) 253 (b) 127 (c) 85 (d) 169
  28. 28.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 400 (c) 202 (d) 301
  29. 29.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 441 (c) 3 (d) 9
  30. 30.If a + b = 16 and ab = 55, what is (a − b)²? (a) 256 (b) 36 (c) 6 (d) 146
  31. 31.If a + b = 14 and ab = 48, what is (a − b)²? (a) 2 (b) 196 (c) 4 (d) 100
  32. 32.What is the sum of the roots of x² − 10x + 21 = 0? (a) 9 (b) 10 (c) 11 (d) 21
  33. 33.What is the sum of the roots of x² − 7x + 10 = 0? (a) 10 (b) 8 (c) 6 (d) 7
  34. 34.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 7569 (b) 8091 (c) 8649 (d) 7579
  35. 35.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 391 (b) 299 (c) 289 (d) 529
  36. 36.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6561 (b) 6399 (c) 6241 (d) 6251
  37. 37.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3604 (b) 3364 (c) 3844 (d) 3854
  38. 38.If a + b = 13 and ab = 12, what is (a − b)²? (a) 11 (b) 169 (c) 145 (d) 121
  39. 39.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)
  40. 40.If a + b = 10 and ab = 21, what is a² + b²? (a) 58 (b) 142 (c) 100 (d) 79

Answer key

  1. 6
  2. 2025
  3. 4761
  4. 74
  5. 4356
  6. 1225
  7. (x + 4)(x + 5)
  8. (x + 4)(x + 8)
  9. 2916
  10. 72
  11. 34
  12. 6 and 7
  13. 4
  14. 49
  15. 25
  16. 225
  17. 32
  18. (x + 2)(x + 8)
  19. 16
  20. 8
  21. 61
  22. 4 and 9
  23. 3 and 9
  24. 185
  25. 64
  26. (x + 2)(x + 7)
  27. 85
  28. 202
  29. 9
  30. 36
  31. 4
  32. 10
  33. 7
  34. 7569
  35. 289
  36. 6241
  37. 3844
  38. 121
  39. (x + 5)(x + 7)
  40. 58

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