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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 401 · 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 − 9x + 20 = 0? (a) 20 (b) 8 (c) 10 (d) 9
  2. 2.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x + 1)(x + 16)
  3. 3.What is the sum of the roots of x² − 12x + 27 = 0? (a) 13 (b) 27 (c) 11 (d) 12
  4. 4.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)
  5. 5.What are the roots of x² − 14x + 45 = 0? (a) 1 and 13 (b) 5 and 9 (c) 3 and 15 (d) 1 and 45
  6. 6.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 62 (c) 16 (d) 63
  7. 7.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 4909 (b) 5329 (c) 4489 (d) 5339
  8. 8.If a + b = 9 and ab = 14, what is (a − b)²? (a) 5 (b) 53 (c) 25 (d) 81
  9. 9.What is the product of the roots of x² − 13x + 40 = 0? (a) 41 (b) 13 (c) 39 (d) 40
  10. 10.What are the roots of x² − 6x + 8 = 0? (a) 2 and 5 (b) 1 and 5 (c) 2 and 4 (d) 1 and 8
  11. 11.If a + b = 15 and ab = 56, what is a² + b²? (a) 113 (b) 225 (c) 337 (d) 169
  12. 12.If a + b = 14 and ab = 45, what is (a − b)²? (a) 4 (b) 16 (c) 196 (d) 106
  13. 13.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1596 (b) 1764 (c) 1444 (d) 1454
  14. 14.If a + b = 9 and ab = 18, what is (a − b)²? (a) 45 (b) 81 (c) 3 (d) 9
  15. 15.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 225 (b) 375 (c) 625 (d) 235
  16. 16.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 409 (b) 529 (c) 539 (d) 289
  17. 17.If a + b = 5 and ab = 4, what is (a − b)²? (a) 25 (b) 3 (c) 17 (d) 9
  18. 18.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1936 (b) 1584 (c) 1306 (d) 1296
  19. 19.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 15) (c) (x + 7)(x + 9) (d) (x + 1)(x + 63)
  20. 20.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 3)(x + 6) (c) (x + 1)(x + 8) (d) (x + 1)(x + 18)
  21. 21.What are the roots of x² − 8x + 12 = 0? (a) 3 and 4 (b) 1 and 12 (c) 1 and 7 (d) 2 and 6
  22. 22.Which is the correct factorisation of x² + 12x + 32? (a) (x + 4)(x + 8) (b) (x − 4)(x − 8) (c) (x + 1)(x + 11) (d) (x + 1)(x + 32)
  23. 23.If a + b = 9 and ab = 14, what is a² + b²? (a) 81 (b) 67 (c) 109 (d) 53
  24. 24.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x + 1)(x + 10) (c) (x − 5)(x − 6) (d) (x + 1)(x + 30)
  25. 25.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 14 (c) 16 (d) 56
  26. 26.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1156 (b) 1166 (c) 676 (d) 916
  27. 27.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 1)(x + 48) (c) (x − 6)(x − 8) (d) (x + 6)(x + 8)
  28. 28.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4901 (b) 4761 (c) 5051 (d) 5041
  29. 29.Which is the correct factorisation of x² + 10x + 21? (a) (x − 3)(x − 7) (b) (x + 3)(x + 7) (c) (x + 1)(x + 21) (d) (x + 1)(x + 9)
  30. 30.What are the roots of x² − 14x + 48 = 0? (a) 2 and 24 (b) 1 and 13 (c) 6 and 8 (d) 1 and 48
  31. 31.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7225 (b) 9025 (c) 7235 (d) 8075
  32. 32.If a + b = 19 and ab = 84, what is (a − b)²? (a) 361 (b) 25 (c) 5 (d) 193
  33. 33.If a + b = 17 and ab = 70, what is (a − b)²? (a) 149 (b) 9 (c) 3 (d) 289
  34. 34.If a + b = 7 and ab = 6, what is (a − b)²? (a) 37 (b) 25 (c) 5 (d) 49
  35. 35.If a + b = 12 and ab = 27, what is (a − b)²? (a) 144 (b) 36 (c) 6 (d) 90
  36. 36.If a + b = 20 and ab = 99, what is a² + b²? (a) 202 (b) 598 (c) 301 (d) 400
  37. 37.What is the sum of the roots of x² − 13x + 42 = 0? (a) 13 (b) 14 (c) 12 (d) 42
  38. 38.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 271 (c) 361 (d) 1
  39. 39.What is the product of the roots of x² − 10x + 16 = 0? (a) 15 (b) 17 (c) 16 (d) 10
  40. 40.If a + b = 14 and ab = 45, what is a² + b²? (a) 196 (b) 151 (c) 286 (d) 106

Answer key

  1. 9
  2. (x + 8)(x + 9)
  3. 12
  4. (x + 2)(x + 7)
  5. 5 and 9
  6. 63
  7. 5329
  8. 25
  9. 40
  10. 2 and 4
  11. 113
  12. 16
  13. 1444
  14. 9
  15. 225
  16. 529
  17. 9
  18. 1296
  19. (x + 7)(x + 9)
  20. (x + 3)(x + 6)
  21. 2 and 6
  22. (x + 4)(x + 8)
  23. 53
  24. (x + 5)(x + 6)
  25. 15
  26. 1156
  27. (x + 6)(x + 8)
  28. 5041
  29. (x + 3)(x + 7)
  30. 6 and 8
  31. 7225
  32. 25
  33. 9
  34. 25
  35. 36
  36. 202
  37. 13
  38. 1
  39. 16
  40. 106

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