← Identities, polynomials and quadratic roots lesson New set →

Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 122 · 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 = 19 and ab = 88, what is (a − b)²? (a) 361 (b) 3 (c) 185 (d) 9
  2. 2.If a + b = 9 and ab = 20, what is a² + b²? (a) 81 (b) 41 (c) 121 (d) 61
  3. 3.If a + b = 17 and ab = 70, what is (a − b)²? (a) 9 (b) 149 (c) 3 (d) 289
  4. 4.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2504 (b) 2304 (c) 2714 (d) 2704
  5. 5.If a + b = 19 and ab = 88, what is a² + b²? (a) 185 (b) 537 (c) 273 (d) 361
  6. 6.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 375 (b) 225 (c) 235 (d) 625
  7. 7.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 2)(x + 4) (c) (x + 1)(x + 5) (d) (x + 1)(x + 8)
  8. 8.What is the sum of the roots of x² − 10x + 21 = 0? (a) 10 (b) 9 (c) 21 (d) 11
  9. 9.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 361 (c) 271 (d) 181
  10. 10.If a + b = 7 and ab = 12, what is (a − b)²? (a) 1 (b) 25 (c) 49 (d) 37
  11. 11.If a + b = 9 and ab = 18, what is (a − b)²? (a) 45 (b) 81 (c) 3 (d) 9
  12. 12.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 5)(x + 7) (c) (x + 1)(x + 35) (d) (x + 1)(x + 11)
  13. 13.If a + b = 15 and ab = 36, what is (a − b)²? (a) 153 (b) 81 (c) 225 (d) 9
  14. 14.If a + b = 15 and ab = 56, what is (a − b)²? (a) 113 (b) 225 (c) 169 (d) 1
  15. 15.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)
  16. 16.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 5)(x + 8) (c) (x + 1)(x + 40) (d) (x + 1)(x + 12)
  17. 17.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3721 (b) 3491 (c) 3481 (d) 3599
  18. 18.If a + b = 17 and ab = 72, what is a² + b²? (a) 145 (b) 433 (c) 289 (d) 217
  19. 19.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 784 (b) 1024 (c) 1034 (d) 904
  20. 20.If a + b = 12 and ab = 27, what is a² + b²? (a) 144 (b) 117 (c) 90 (d) 198
  21. 21.If a + b = 10 and ab = 24, what is (a − b)²? (a) 52 (b) 100 (c) 2 (d) 4
  22. 22.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 739 (b) 1089 (c) 729 (d) 891
  23. 23.What is the sum of the roots of x² − 11x + 24 = 0? (a) 24 (b) 12 (c) 11 (d) 10
  24. 24.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4904 (b) 5184 (c) 4624 (d) 5194
  25. 25.If a + b = 19 and ab = 84, what is (a − b)²? (a) 25 (b) 5 (c) 193 (d) 361
  26. 26.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 15) (c) (x + 1)(x + 63) (d) (x + 7)(x + 9)
  27. 27.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x + 1)(x + 13) (c) (x − 6)(x − 8) (d) (x + 1)(x + 48)
  28. 28.If a + b = 18 and ab = 77, what is (a − b)²? (a) 4 (b) 324 (c) 170 (d) 16
  29. 29.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2209 (b) 2491 (c) 2809 (d) 2219
  30. 30.If a + b = 17 and ab = 72, what is (a − b)²? (a) 217 (b) 1 (c) 145 (d) 289
  31. 31.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)
  32. 32.What is the sum of the roots of x² − 13x + 42 = 0? (a) 42 (b) 14 (c) 12 (d) 13
  33. 33.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 71 (c) 17 (d) 73
  34. 34.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3035 (b) 4225 (c) 3025 (d) 3575
  35. 35.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 425 (b) 225 (c) 635 (d) 625
  36. 36.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 55 (c) 56 (d) 57
  37. 37.If a + b = 12 and ab = 32, what is a² + b²? (a) 208 (b) 144 (c) 80 (d) 112
  38. 38.What is the product of the roots of x² − 13x + 42 = 0? (a) 43 (b) 13 (c) 41 (d) 42
  39. 39.What is the product of the roots of x² − 7x + 12 = 0? (a) 11 (b) 12 (c) 7 (d) 13
  40. 40.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2525 (b) 3025 (c) 3035 (d) 2025

Answer key

  1. 9
  2. 41
  3. 9
  4. 2704
  5. 185
  6. 225
  7. (x + 2)(x + 4)
  8. 10
  9. 181
  10. 1
  11. 9
  12. (x + 5)(x + 7)
  13. 81
  14. 1
  15. (x + 6)(x + 7)
  16. (x + 5)(x + 8)
  17. 3481
  18. 145
  19. 1024
  20. 90
  21. 4
  22. 729
  23. 11
  24. 5184
  25. 25
  26. (x + 7)(x + 9)
  27. (x + 6)(x + 8)
  28. 16
  29. 2209
  30. 1
  31. (x + 7)(x + 8)
  32. 13
  33. 72
  34. 3025
  35. 625
  36. 56
  37. 80
  38. 42
  39. 12
  40. 3025

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