← Identities, polynomials and quadratic roots lesson New set →

Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 816 · 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 a² + b²? (a) 273 (b) 361 (c) 185 (d) 537
  2. 2.What is the sum of the roots of x² − 10x + 24 = 0? (a) 9 (b) 10 (c) 11 (d) 24
  3. 3.If a + b = 11 and ab = 30, what is (a − b)²? (a) 1 (b) 61 (c) 121 (d) 91
  4. 4.What is the product of the roots of x² − 13x + 40 = 0? (a) 13 (b) 41 (c) 39 (d) 40
  5. 5.What is the sum of the roots of x² − 10x + 16 = 0? (a) 11 (b) 9 (c) 16 (d) 10
  6. 6.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 202 (c) 400 (d) 4
  7. 7.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1936 (b) 1584 (c) 1306 (d) 1296
  8. 8.What is the product of the roots of x² − 13x + 42 = 0? (a) 41 (b) 43 (c) 13 (d) 42
  9. 9.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 7)(x + 9) (c) (x + 1)(x + 15) (d) (x + 1)(x + 63)
  10. 10.What is the sum of the roots of x² − 8x + 15 = 0? (a) 7 (b) 15 (c) 9 (d) 8
  11. 11.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1531 (b) 1681 (c) 1521 (d) 1599
  12. 12.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x − 5)(x − 7) (c) (x + 5)(x + 7) (d) (x + 1)(x + 11)
  13. 13.If a + b = 15 and ab = 44, what is a² + b²? (a) 225 (b) 313 (c) 181 (d) 137
  14. 14.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 324 (b) 494 (c) 404 (d) 484
  15. 15.If a + b = 8 and ab = 7, what is a² + b²? (a) 78 (b) 64 (c) 50 (d) 57
  16. 16.If a + b = 7 and ab = 10, what is (a − b)²? (a) 29 (b) 49 (c) 3 (d) 9
  17. 17.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 62 (c) 63 (d) 16
  18. 18.If a + b = 12 and ab = 35, what is a² + b²? (a) 144 (b) 109 (c) 74 (d) 214
  19. 19.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1099 (b) 909 (c) 1089 (d) 729
  20. 20.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 72) (c) (x − 8)(x − 9) (d) (x + 1)(x + 16)
  21. 21.If a + b = 20 and ab = 96, what is (a − b)²? (a) 208 (b) 400 (c) 4 (d) 16
  22. 22.If a + b = 16 and ab = 55, what is (a − b)²? (a) 6 (b) 256 (c) 146 (d) 36
  23. 23.If a + b = 14 and ab = 33, what is a² + b²? (a) 262 (b) 196 (c) 130 (d) 163
  24. 24.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x − 6)(x − 9) (c) (x + 1)(x + 54) (d) (x + 1)(x + 14)
  25. 25.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3854 (b) 3364 (c) 3604 (d) 3844
  26. 26.If a + b = 17 and ab = 60, what is (a − b)²? (a) 49 (b) 289 (c) 169 (d) 7
  27. 27.If a + b = 9 and ab = 18, what is a² + b²? (a) 117 (b) 45 (c) 81 (d) 63
  28. 28.If a + b = 9 and ab = 20, what is a² + b²? (a) 61 (b) 81 (c) 41 (d) 121
  29. 29.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 7569 (b) 7579 (c) 8091 (d) 8649
  30. 30.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4634 (b) 4624 (c) 4896 (d) 5184
  31. 31.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 1 (c) 271 (d) 361
  32. 32.If a + b = 5 and ab = 4, what is (a − b)²? (a) 9 (b) 3 (c) 17 (d) 25
  33. 33.If a + b = 7 and ab = 10, what is a² + b²? (a) 49 (b) 69 (c) 29 (d) 39
  34. 34.What is the sum of the roots of x² − 6x + 8 = 0? (a) 7 (b) 5 (c) 6 (d) 8
  35. 35.What are the roots of x² − 16x + 63 = 0? (a) 1 and 63 (b) 7 and 9 (c) 1 and 15 (d) 3 and 21
  36. 36.If a + b = 9 and ab = 8, what is a² + b²? (a) 73 (b) 81 (c) 65 (d) 97
  37. 37.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 56) (c) (x + 1)(x + 14) (d) (x + 7)(x + 8)
  38. 38.Which is the correct factorisation of x² + 13x + 40? (a) (x + 5)(x + 8) (b) (x + 1)(x + 12) (c) (x − 5)(x − 8) (d) (x + 1)(x + 40)
  39. 39.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2304 (b) 2704 (c) 2504 (d) 2714
  40. 40.What are the roots of x² − 7x + 12 = 0? (a) 3 and 4 (b) 1 and 12 (c) 1 and 6 (d) 2 and 6

Answer key

  1. 185
  2. 10
  3. 1
  4. 40
  5. 10
  6. 4
  7. 1296
  8. 42
  9. (x + 7)(x + 9)
  10. 8
  11. 1521
  12. (x + 5)(x + 7)
  13. 137
  14. 484
  15. 50
  16. 9
  17. 63
  18. 74
  19. 1089
  20. (x + 8)(x + 9)
  21. 16
  22. 36
  23. 130
  24. (x + 6)(x + 9)
  25. 3844
  26. 49
  27. 45
  28. 41
  29. 7569
  30. 4624
  31. 1
  32. 9
  33. 29
  34. 6
  35. 7 and 9
  36. 65
  37. (x + 7)(x + 8)
  38. (x + 5)(x + 8)
  39. 2704
  40. 3 and 4

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