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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 47 · 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 + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 14) (c) (x + 1)(x + 56) (d) (x + 7)(x + 8)
  2. 2.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x + 2)(x + 6) (c) (x − 2)(x − 6) (d) (x + 1)(x + 12)
  3. 3.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4761 (b) 5041 (c) 4901 (d) 5051
  4. 4.Which is the correct factorisation of x² + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 1)(x + 16) (c) (x + 2)(x + 8) (d) (x + 1)(x + 9)
  5. 5.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 494 (b) 484 (c) 324 (d) 404
  6. 6.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 10201 (b) 9811 (c) 9999 (d) 9801
  7. 7.If a + b = 11 and ab = 18, what is (a − b)²? (a) 121 (b) 85 (c) 49 (d) 7
  8. 8.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 1)(x + 72)
  9. 9.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 18 (c) 16 (d) 72
  10. 10.What is the product of the roots of x² − 11x + 24 = 0? (a) 25 (b) 11 (c) 24 (d) 23
  11. 11.If a + b = 15 and ab = 36, what is (a − b)²? (a) 9 (b) 81 (c) 153 (d) 225
  12. 12.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4356 (b) 5476 (c) 4916 (d) 5486
  13. 13.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 1)(x + 5) (c) (x + 1)(x + 8) (d) (x + 2)(x + 4)
  14. 14.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x − 4)(x − 6) (c) (x + 4)(x + 6) (d) (x + 1)(x + 9)
  15. 15.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x + 4)(x + 8) (c) (x + 1)(x + 11) (d) (x − 4)(x − 8)
  16. 16.What is the sum of the roots of x² − 5x + 6 = 0? (a) 5 (b) 6 (c) 7 (d) 4
  17. 17.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x − 4)(x − 7) (c) (x + 4)(x + 7) (d) (x + 1)(x + 28)
  18. 18.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3249 (b) 3591 (c) 3969 (d) 3259
  19. 19.If a + b = 20 and ab = 96, what is a² + b²? (a) 304 (b) 400 (c) 592 (d) 208
  20. 20.What is the product of the roots of x² − 12x + 32 = 0? (a) 33 (b) 12 (c) 31 (d) 32
  21. 21.If a + b = 20 and ab = 99, what is a² + b²? (a) 301 (b) 202 (c) 598 (d) 400
  22. 22.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1369 (b) 1609 (c) 1859 (d) 1849
  23. 23.If a + b = 8 and ab = 15, what is a² + b²? (a) 94 (b) 64 (c) 49 (d) 34
  24. 24.If a + b = 9 and ab = 20, what is a² + b²? (a) 41 (b) 61 (c) 81 (d) 121
  25. 25.If a + b = 13 and ab = 42, what is (a − b)²? (a) 169 (b) 85 (c) 1 (d) 127
  26. 26.If a + b = 12 and ab = 35, what is a² + b²? (a) 109 (b) 214 (c) 74 (d) 144
  27. 27.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 7744 (b) 8464 (c) 8104 (d) 8474
  28. 28.If a + b = 16 and ab = 63, what is (a − b)²? (a) 256 (b) 130 (c) 4 (d) 2
  29. 29.If a + b = 18 and ab = 80, what is a² + b²? (a) 244 (b) 484 (c) 164 (d) 324
  30. 30.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2601 (b) 2501 (c) 2611 (d) 2401
  31. 31.If a + b = 10 and ab = 16, what is a² + b²? (a) 84 (b) 68 (c) 100 (d) 132
  32. 32.If a + b = 16 and ab = 55, what is (a − b)²? (a) 256 (b) 146 (c) 6 (d) 36
  33. 33.If a + b = 20 and ab = 99, what is (a − b)²? (a) 4 (b) 2 (c) 202 (d) 400
  34. 34.If a + b = 12 and ab = 11, what is (a − b)²? (a) 10 (b) 100 (c) 144 (d) 122
  35. 35.What are the roots of x² − 11x + 30 = 0? (a) 1 and 30 (b) 5 and 6 (c) 1 and 10 (d) 2 and 15
  36. 36.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6396 (b) 6094 (c) 6724 (d) 6084
  37. 37.What are the roots of x² − 8x + 15 = 0? (a) 2 and 6 (b) 3 and 5 (c) 1 and 7 (d) 1 and 15
  38. 38.If a + b = 17 and ab = 70, what is (a − b)²? (a) 289 (b) 9 (c) 3 (d) 149
  39. 39.If a + b = 15 and ab = 50, what is (a − b)²? (a) 5 (b) 125 (c) 225 (d) 25
  40. 40.If a + b = 17 and ab = 72, what is a² + b²? (a) 145 (b) 433 (c) 217 (d) 289

Answer key

  1. (x + 7)(x + 8)
  2. (x + 2)(x + 6)
  3. 5041
  4. (x + 2)(x + 8)
  5. 484
  6. 9801
  7. 49
  8. (x + 8)(x + 9)
  9. 17
  10. 24
  11. 81
  12. 5476
  13. (x + 2)(x + 4)
  14. (x + 4)(x + 6)
  15. (x + 4)(x + 8)
  16. 5
  17. (x + 4)(x + 7)
  18. 3249
  19. 208
  20. 32
  21. 202
  22. 1849
  23. 34
  24. 41
  25. 1
  26. 74
  27. 8464
  28. 4
  29. 164
  30. 2601
  31. 68
  32. 36
  33. 4
  34. 100
  35. 5 and 6
  36. 6084
  37. 3 and 5
  38. 9
  39. 25
  40. 145

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