← Identities, polynomials and quadratic roots lesson New set →

Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 477 · 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 = 11 and ab = 10, what is a² + b²? (a) 101 (b) 121 (c) 111 (d) 141
  2. 2.If a + b = 18 and ab = 80, what is (a − b)²? (a) 324 (b) 164 (c) 2 (d) 4
  3. 3.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 1)(x + 15) (c) (x + 7)(x + 9) (d) (x − 7)(x − 9)
  4. 4.If a + b = 15 and ab = 44, what is a² + b²? (a) 313 (b) 181 (c) 225 (d) 137
  5. 5.What are the roots of x² − 16x + 63 = 0? (a) 3 and 21 (b) 1 and 15 (c) 1 and 63 (d) 7 and 9
  6. 6.If a + b = 9 and ab = 14, what is a² + b²? (a) 109 (b) 53 (c) 67 (d) 81
  7. 7.If a + b = 19 and ab = 90, what is (a − b)²? (a) 1 (b) 271 (c) 361 (d) 181
  8. 8.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2126 (b) 2484 (c) 2916 (d) 2116
  9. 9.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 1)(x + 14) (c) (x + 6)(x + 9) (d) (x − 6)(x − 9)
  10. 10.If a + b = 16 and ab = 63, what is a² + b²? (a) 193 (b) 130 (c) 256 (d) 382
  11. 11.If a + b = 7 and ab = 12, what is a² + b²? (a) 49 (b) 25 (c) 37 (d) 73
  12. 12.What is the sum of the roots of x² − 11x + 24 = 0? (a) 10 (b) 24 (c) 11 (d) 12
  13. 13.Which is the correct factorisation of x² + 13x + 36? (a) (x + 4)(x + 9) (b) (x + 1)(x + 36) (c) (x − 4)(x − 9) (d) (x + 1)(x + 12)
  14. 14.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10004 (b) 10404 (c) 10414 (d) 9604
  15. 15.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 324 (b) 404 (c) 494 (d) 484
  16. 16.If a + b = 5 and ab = 6, what is a² + b²? (a) 19 (b) 37 (c) 25 (d) 13
  17. 17.If a + b = 11 and ab = 28, what is a² + b²? (a) 121 (b) 177 (c) 93 (d) 65
  18. 18.If a + b = 6 and ab = 5, what is (a − b)²? (a) 16 (b) 4 (c) 26 (d) 36
  19. 19.If a + b = 20 and ab = 96, what is (a − b)²? (a) 4 (b) 16 (c) 208 (d) 400
  20. 20.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 1 and 16 (c) 1 and 72 (d) 2 and 36
  21. 21.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 18 (c) 72 (d) 17
  22. 22.If a + b = 12 and ab = 27, what is a² + b²? (a) 90 (b) 144 (c) 117 (d) 198
  23. 23.What is the sum of the roots of x² − 8x + 12 = 0? (a) 8 (b) 7 (c) 9 (d) 12
  24. 24.If a + b = 9 and ab = 18, what is a² + b²? (a) 63 (b) 81 (c) 45 (d) 117
  25. 25.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
  26. 26.Which is the correct factorisation of x² + 5x + 6? (a) (x + 2)(x + 3) (b) (x − 2)(x − 3) (c) (x + 1)(x + 4) (d) (x + 1)(x + 6)
  27. 27.If a + b = 8 and ab = 12, what is a² + b²? (a) 64 (b) 88 (c) 40 (d) 52
  28. 28.What is the sum of the roots of x² − 9x + 20 = 0? (a) 10 (b) 9 (c) 8 (d) 20
  29. 29.What is the product of the roots of x² − 15x + 54 = 0? (a) 53 (b) 54 (c) 15 (d) 55
  30. 30.What is the sum of the roots of x² − 11x + 30 = 0? (a) 12 (b) 11 (c) 30 (d) 10
  31. 31.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 256 (b) 576 (c) 384 (d) 266
  32. 32.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)
  33. 33.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 3)(x + 5) (c) (x + 1)(x + 15) (d) (x + 1)(x + 7)
  34. 34.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 334 (b) 396 (c) 484 (d) 324
  35. 35.Which is the correct factorisation of x² + 9x + 14? (a) (x + 2)(x + 7) (b) (x − 2)(x − 7) (c) (x + 1)(x + 14) (d) (x + 1)(x + 8)
  36. 36.If a + b = 10 and ab = 16, what is (a − b)²? (a) 6 (b) 68 (c) 100 (d) 36
  37. 37.What are the roots of x² − 13x + 42 = 0? (a) 6 and 7 (b) 1 and 42 (c) 2 and 21 (d) 1 and 12
  38. 38.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 4 (c) 400 (d) 202
  39. 39.What is the product of the roots of x² − 8x + 12 = 0? (a) 11 (b) 13 (c) 12 (d) 8
  40. 40.If a + b = 17 and ab = 60, what is a² + b²? (a) 409 (b) 289 (c) 229 (d) 169

Answer key

  1. 101
  2. 4
  3. (x + 7)(x + 9)
  4. 137
  5. 7 and 9
  6. 53
  7. 1
  8. 2116
  9. (x + 6)(x + 9)
  10. 130
  11. 25
  12. 11
  13. (x + 4)(x + 9)
  14. 10404
  15. 484
  16. 13
  17. 65
  18. 16
  19. 16
  20. 8 and 9
  21. 17
  22. 90
  23. 8
  24. 45
  25. (x + 8)(x + 9)
  26. (x + 2)(x + 3)
  27. 40
  28. 9
  29. 54
  30. 11
  31. 256
  32. (x + 2)(x + 8)
  33. (x + 3)(x + 5)
  34. 324
  35. (x + 2)(x + 7)
  36. 36
  37. 6 and 7
  38. 4
  39. 12
  40. 169

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