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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 670 · 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.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 13 (c) 14 (d) 15
  2. 2.If a + b = 6 and ab = 8, what is a² + b²? (a) 28 (b) 52 (c) 20 (d) 36
  3. 3.What is the sum of the roots of x² − 10x + 24 = 0? (a) 24 (b) 11 (c) 9 (d) 10
  4. 4.What is the sum of the roots of x² − 6x + 8 = 0? (a) 6 (b) 8 (c) 7 (d) 5
  5. 5.If a + b = 17 and ab = 66, what is a² + b²? (a) 421 (b) 223 (c) 289 (d) 157
  6. 6.What are the roots of x² − 14x + 48 = 0? (a) 2 and 24 (b) 1 and 48 (c) 1 and 13 (d) 6 and 8
  7. 7.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 16 (c) 72 (d) 18
  8. 8.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1454 (b) 1444 (c) 1596 (d) 1764
  9. 9.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 1)(x + 11) (c) (x + 5)(x + 7) (d) (x + 1)(x + 35)
  10. 10.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 484 (b) 334 (c) 396 (d) 324
  11. 11.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 17 (c) 72 (d) 71
  12. 12.What are the roots of x² − 10x + 24 = 0? (a) 1 and 9 (b) 2 and 12 (c) 4 and 6 (d) 1 and 24
  13. 13.If a + b = 3 and ab = 2, what is a² + b²? (a) 7 (b) 13 (c) 9 (d) 5
  14. 14.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5051 (b) 4901 (c) 4761 (d) 5041
  15. 15.If a + b = 10 and ab = 9, what is (a − b)²? (a) 82 (b) 64 (c) 100 (d) 8
  16. 16.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 7)(x + 9) (c) (x + 1)(x + 63) (d) (x − 7)(x − 9)
  17. 17.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 375 (b) 225 (c) 625 (d) 235
  18. 18.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 529 (b) 539 (c) 289 (d) 409
  19. 19.If a + b = 21 and ab = 108, what is a² + b²? (a) 225 (b) 657 (c) 441 (d) 333
  20. 20.If a + b = 15 and ab = 56, what is (a − b)²? (a) 225 (b) 1 (c) 113 (d) 169
  21. 21.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 9025 (b) 7225 (c) 8125 (d) 9035
  22. 22.If a + b = 13 and ab = 42, what is (a − b)²? (a) 127 (b) 1 (c) 169 (d) 85
  23. 23.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)
  24. 24.What is the sum of the roots of x² − 8x + 15 = 0? (a) 8 (b) 7 (c) 9 (d) 15
  25. 25.If a + b = 19 and ab = 90, what is (a − b)²? (a) 361 (b) 1 (c) 181 (d) 271
  26. 26.If a + b = 12 and ab = 32, what is (a − b)²? (a) 144 (b) 16 (c) 80 (d) 4
  27. 27.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 961 (b) 901 (c) 841 (d) 971
  28. 28.If a + b = 12 and ab = 32, what is a² + b²? (a) 208 (b) 144 (c) 80 (d) 112
  29. 29.What is the product of the roots of x² − 13x + 42 = 0? (a) 13 (b) 41 (c) 42 (d) 43
  30. 30.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 10201 (b) 9811 (c) 9999 (d) 9801
  31. 31.What are the roots of x² − 13x + 42 = 0? (a) 2 and 21 (b) 6 and 7 (c) 1 and 12 (d) 1 and 42
  32. 32.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x − 5)(x − 9) (c) (x + 5)(x + 9) (d) (x + 1)(x + 45)
  33. 33.If a + b = 14 and ab = 24, what is a² + b²? (a) 196 (b) 172 (c) 244 (d) 148
  34. 34.If a + b = 11 and ab = 30, what is (a − b)²? (a) 61 (b) 91 (c) 1 (d) 121
  35. 35.What is the product of the roots of x² − 6x + 8 = 0? (a) 6 (b) 7 (c) 9 (d) 8
  36. 36.If a + b = 19 and ab = 88, what is a² + b²? (a) 273 (b) 537 (c) 185 (d) 361
  37. 37.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3374 (b) 3844 (c) 3596 (d) 3364
  38. 38.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 1)(x + 14) (c) (x + 2)(x + 7) (d) (x + 1)(x + 8)
  39. 39.If a + b = 15 and ab = 36, what is a² + b²? (a) 189 (b) 225 (c) 153 (d) 297
  40. 40.If a + b = 15 and ab = 56, what is a² + b²? (a) 113 (b) 337 (c) 225 (d) 169

Answer key

  1. 14
  2. 20
  3. 10
  4. 6
  5. 157
  6. 6 and 8
  7. 17
  8. 1444
  9. (x + 5)(x + 7)
  10. 324
  11. 72
  12. 4 and 6
  13. 5
  14. 5041
  15. 64
  16. (x + 7)(x + 9)
  17. 225
  18. 529
  19. 225
  20. 1
  21. 9025
  22. 1
  23. (x + 8)(x + 9)
  24. 8
  25. 1
  26. 16
  27. 961
  28. 80
  29. 42
  30. 9801
  31. 6 and 7
  32. (x + 5)(x + 9)
  33. 148
  34. 1
  35. 8
  36. 185
  37. 3364
  38. (x + 2)(x + 7)
  39. 153
  40. 113

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