← Identities, polynomials and quadratic roots lesson New set →

Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 549 · 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 + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 1)(x + 14) (c) (x + 1)(x + 8) (d) (x + 2)(x + 7)
  2. 2.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1774 (b) 1604 (c) 1764 (d) 1444
  3. 3.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7754 (b) 7744 (c) 8464 (d) 8096
  4. 4.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 4)(x + 6) (c) (x + 1)(x + 9) (d) (x + 1)(x + 24)
  5. 5.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 7744 (b) 8464 (c) 8474 (d) 8104
  6. 6.What is the sum of the roots of x² − 15x + 54 = 0? (a) 15 (b) 16 (c) 54 (d) 14
  7. 7.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2504 (b) 2304 (c) 2704 (d) 2714
  8. 8.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x − 3)(x − 4) (c) (x + 3)(x + 4) (d) (x + 1)(x + 12)
  9. 9.If a + b = 18 and ab = 77, what is a² + b²? (a) 478 (b) 247 (c) 324 (d) 170
  10. 10.What are the roots of x² − 9x + 14 = 0? (a) 1 and 14 (b) 2 and 7 (c) 1 and 8 (d) 3 and 6
  11. 11.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 2 and 36 (c) 1 and 72 (d) 8 and 9
  12. 12.What is the sum of the roots of x² − 9x + 18 = 0? (a) 10 (b) 9 (c) 8 (d) 18
  13. 13.If a + b = 21 and ab = 108, what is (a − b)²? (a) 9 (b) 441 (c) 225 (d) 3
  14. 14.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 409 (b) 539 (c) 289 (d) 529
  15. 15.What is the product of the roots of x² − 16x + 63 = 0? (a) 16 (b) 64 (c) 62 (d) 63
  16. 16.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 225 (b) 635 (c) 425 (d) 625
  17. 17.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 15 (c) 16 (d) 17
  18. 18.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 1225 (b) 875 (c) 625 (d) 635
  19. 19.If a + b = 13 and ab = 40, what is a² + b²? (a) 129 (b) 89 (c) 249 (d) 169
  20. 20.What is the sum of the roots of x² − 10x + 24 = 0? (a) 24 (b) 11 (c) 9 (d) 10
  21. 21.If a + b = 13 and ab = 22, what is (a − b)²? (a) 9 (b) 169 (c) 125 (d) 81
  22. 22.If a + b = 11 and ab = 10, what is (a − b)²? (a) 81 (b) 9 (c) 121 (d) 101
  23. 23.If a + b = 20 and ab = 96, what is a² + b²? (a) 592 (b) 208 (c) 304 (d) 400
  24. 24.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4875 (b) 5625 (c) 4235 (d) 4225
  25. 25.If a + b = 12 and ab = 27, what is (a − b)²? (a) 36 (b) 90 (c) 6 (d) 144
  26. 26.If a + b = 10 and ab = 21, what is a² + b²? (a) 79 (b) 142 (c) 58 (d) 100
  27. 27.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x + 1)(x + 14) (c) (x + 1)(x + 54) (d) (x − 6)(x − 9)
  28. 28.What is the product of the roots of x² − 12x + 35 = 0? (a) 34 (b) 35 (c) 12 (d) 36
  29. 29.If a + b = 19 and ab = 88, what is a² + b²? (a) 361 (b) 537 (c) 273 (d) 185
  30. 30.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 686 (b) 884 (c) 1156 (d) 676
  31. 31.If a + b = 12 and ab = 32, what is a² + b²? (a) 112 (b) 144 (c) 208 (d) 80
  32. 32.If a + b = 9 and ab = 14, what is (a − b)²? (a) 25 (b) 53 (c) 81 (d) 5
  33. 33.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
  34. 34.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2611 (b) 2401 (c) 2501 (d) 2601
  35. 35.What is the product of the roots of x² − 13x + 36 = 0? (a) 13 (b) 35 (c) 36 (d) 37
  36. 36.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 73 (c) 72 (d) 71
  37. 37.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3374 (b) 3596 (c) 3364 (d) 3844
  38. 38.If a + b = 9 and ab = 20, what is (a − b)²? (a) 41 (b) 81 (c) 61 (d) 1
  39. 39.If a + b = 12 and ab = 35, what is (a − b)²? (a) 4 (b) 2 (c) 144 (d) 74
  40. 40.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 361 (c) 271 (d) 1

Answer key

  1. (x + 2)(x + 7)
  2. 1764
  3. 7744
  4. (x + 4)(x + 6)
  5. 8464
  6. 15
  7. 2704
  8. (x + 3)(x + 4)
  9. 170
  10. 2 and 7
  11. 8 and 9
  12. 9
  13. 9
  14. 529
  15. 63
  16. 625
  17. 16
  18. 625
  19. 89
  20. 10
  21. 81
  22. 81
  23. 208
  24. 4225
  25. 36
  26. 58
  27. (x + 6)(x + 9)
  28. 35
  29. 185
  30. 676
  31. 80
  32. 25
  33. (x + 8)(x + 9)
  34. 2601
  35. 36
  36. 72
  37. 3364
  38. 1
  39. 4
  40. 1

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