← Identities, polynomials and quadratic roots lesson New set →

Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 346 · 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.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 42) (c) (x + 6)(x + 7) (d) (x + 1)(x + 12)
  2. 2.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2209 (b) 2491 (c) 2809 (d) 2219
  3. 3.If a + b = 16 and ab = 48, what is a² + b²? (a) 352 (b) 160 (c) 256 (d) 208
  4. 4.If a + b = 5 and ab = 4, what is (a − b)²? (a) 9 (b) 3 (c) 17 (d) 25
  5. 5.If a + b = 18 and ab = 80, what is a² + b²? (a) 324 (b) 244 (c) 164 (d) 484
  6. 6.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2501 (b) 2401 (c) 2601 (d) 2611
  7. 7.What are the roots of x² − 10x + 21 = 0? (a) 1 and 9 (b) 3 and 7 (c) 1 and 21 (d) 2 and 8
  8. 8.If a + b = 12 and ab = 35, what is (a − b)²? (a) 4 (b) 144 (c) 2 (d) 74
  9. 9.If a + b = 18 and ab = 80, what is (a − b)²? (a) 164 (b) 324 (c) 2 (d) 4
  10. 10.If a + b = 7 and ab = 6, what is a² + b²? (a) 37 (b) 61 (c) 49 (d) 43
  11. 11.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 5)(x + 6) (c) (x + 1)(x + 30) (d) (x + 1)(x + 10)
  12. 12.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)
  13. 13.If a + b = 8 and ab = 15, what is a² + b²? (a) 94 (b) 64 (c) 34 (d) 49
  14. 14.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10414 (b) 9604 (c) 10004 (d) 10404
  15. 15.If a + b = 14 and ab = 45, what is (a − b)²? (a) 106 (b) 196 (c) 4 (d) 16
  16. 16.What is the product of the roots of x² − 9x + 20 = 0? (a) 19 (b) 20 (c) 9 (d) 21
  17. 17.What are the roots of x² − 16x + 63 = 0? (a) 1 and 15 (b) 3 and 21 (c) 7 and 9 (d) 1 and 63
  18. 18.If a + b = 16 and ab = 48, what is (a − b)²? (a) 160 (b) 8 (c) 256 (d) 64
  19. 19.If a + b = 16 and ab = 63, what is (a − b)²? (a) 2 (b) 4 (c) 130 (d) 256
  20. 20.If a + b = 15 and ab = 56, what is a² + b²? (a) 225 (b) 169 (c) 337 (d) 113
  21. 21.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)
  22. 22.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 4 (c) 202 (d) 2
  23. 23.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1849 (b) 1591 (c) 1369 (d) 1379
  24. 24.If a + b = 17 and ab = 60, what is a² + b²? (a) 409 (b) 289 (c) 169 (d) 229
  25. 25.If a + b = 11 and ab = 30, what is a² + b²? (a) 91 (b) 61 (c) 181 (d) 121
  26. 26.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 3025 (b) 2475 (c) 2035 (d) 2025
  27. 27.What is the sum of the roots of x² − 9x + 14 = 0? (a) 8 (b) 10 (c) 9 (d) 14
  28. 28.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 494 (b) 324 (c) 404 (d) 484
  29. 29.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9801 (b) 9811 (c) 10201 (d) 9999
  30. 30.Which is the correct factorisation of x² + 11x + 24? (a) (x − 3)(x − 8) (b) (x + 3)(x + 8) (c) (x + 1)(x + 24) (d) (x + 1)(x + 10)
  31. 31.What are the roots of x² − 12x + 35 = 0? (a) 1 and 35 (b) 1 and 11 (c) 5 and 7 (d) 2 and 10
  32. 32.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 266 (b) 256 (c) 576 (d) 384
  33. 33.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x + 1)(x + 7) (c) (x + 3)(x + 5) (d) (x − 3)(x − 5)
  34. 34.What is the sum of the roots of x² − 13x + 36 = 0? (a) 13 (b) 36 (c) 12 (d) 14
  35. 35.Which is the correct factorisation of x² + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 2)(x + 8) (c) (x + 1)(x + 16) (d) (x + 1)(x + 9)
  36. 36.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3625 (b) 4235 (c) 4225 (d) 3025
  37. 37.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1521 (b) 1681 (c) 1531 (d) 1599
  38. 38.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4489 (b) 4499 (c) 5329 (d) 4891
  39. 39.If a + b = 11 and ab = 24, what is a² + b²? (a) 169 (b) 73 (c) 97 (d) 121
  40. 40.If a + b = 19 and ab = 84, what is a² + b²? (a) 529 (b) 193 (c) 277 (d) 361

Answer key

  1. (x + 6)(x + 7)
  2. 2209
  3. 160
  4. 9
  5. 164
  6. 2601
  7. 3 and 7
  8. 4
  9. 4
  10. 37
  11. (x + 5)(x + 6)
  12. (x + 3)(x + 9)
  13. 34
  14. 10404
  15. 16
  16. 20
  17. 7 and 9
  18. 64
  19. 4
  20. 113
  21. (x + 7)(x + 8)
  22. 4
  23. 1369
  24. 169
  25. 61
  26. 2025
  27. 9
  28. 484
  29. 9801
  30. (x + 3)(x + 8)
  31. 5 and 7
  32. 256
  33. (x + 3)(x + 5)
  34. 13
  35. (x + 2)(x + 8)
  36. 4225
  37. 1521
  38. 4489
  39. 73
  40. 193

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