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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 897 · 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 are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 1 and 72 (c) 8 and 9 (d) 2 and 36
  2. 2.What is the sum of the roots of x² − 15x + 54 = 0? (a) 14 (b) 16 (c) 54 (d) 15
  3. 3.If a + b = 18 and ab = 77, what is a² + b²? (a) 324 (b) 170 (c) 478 (d) 247
  4. 4.If a + b = 18 and ab = 77, what is (a − b)²? (a) 4 (b) 16 (c) 324 (d) 170
  5. 5.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6889 (b) 6391 (c) 5939 (d) 5929
  6. 6.Which is the correct factorisation of x² + 8x + 12? (a) (x + 2)(x + 6) (b) (x + 1)(x + 12) (c) (x − 2)(x − 6) (d) (x + 1)(x + 7)
  7. 7.What are the roots of x² − 14x + 48 = 0? (a) 1 and 13 (b) 6 and 8 (c) 2 and 24 (d) 1 and 48
  8. 8.What are the roots of x² − 7x + 10 = 0? (a) 3 and 4 (b) 1 and 6 (c) 2 and 5 (d) 1 and 10
  9. 9.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2704 (b) 2304 (c) 2714 (d) 2504
  10. 10.If a + b = 15 and ab = 50, what is a² + b²? (a) 325 (b) 225 (c) 175 (d) 125
  11. 11.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 9 (c) 3 (d) 441
  12. 12.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 1)(x + 10) (c) (x − 5)(x − 6) (d) (x + 5)(x + 6)
  13. 13.If a + b = 15 and ab = 44, what is (a − b)²? (a) 225 (b) 49 (c) 137 (d) 7
  14. 14.What is the product of the roots of x² − 10x + 16 = 0? (a) 10 (b) 17 (c) 16 (d) 15
  15. 15.What is the sum of the roots of x² − 10x + 24 = 0? (a) 11 (b) 10 (c) 9 (d) 24
  16. 16.If a + b = 13 and ab = 22, what is (a − b)²? (a) 9 (b) 81 (c) 125 (d) 169
  17. 17.If a + b = 15 and ab = 56, what is a² + b²? (a) 337 (b) 225 (c) 113 (d) 169
  18. 18.What is the sum of the roots of x² − 12x + 32 = 0? (a) 13 (b) 32 (c) 12 (d) 11
  19. 19.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 416 (b) 256 (c) 586 (d) 576
  20. 20.What are the roots of x² − 11x + 24 = 0? (a) 1 and 10 (b) 2 and 12 (c) 3 and 8 (d) 1 and 24
  21. 21.If a + b = 7 and ab = 6, what is a² + b²? (a) 61 (b) 37 (c) 43 (d) 49
  22. 22.If a + b = 19 and ab = 84, what is (a − b)²? (a) 5 (b) 193 (c) 25 (d) 361
  23. 23.Which is the correct factorisation of x² + 6x + 8? (a) (x + 2)(x + 4) (b) (x + 1)(x + 8) (c) (x − 2)(x − 4) (d) (x + 1)(x + 5)
  24. 24.If a + b = 3 and ab = 2, what is (a − b)²? (a) 5 (b) 1 (c) 7 (d) 9
  25. 25.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 451 (b) 401 (c) 361 (d) 441
  26. 26.What is the sum of the roots of x² − 8x + 15 = 0? (a) 7 (b) 8 (c) 9 (d) 15
  27. 27.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 4225 (b) 3575 (c) 3035 (d) 3025
  28. 28.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)
  29. 29.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 1024 (b) 784 (c) 794 (d) 896
  30. 30.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2314 (b) 2304 (c) 2704 (d) 2496
  31. 31.If a + b = 7 and ab = 12, what is (a − b)²? (a) 25 (b) 49 (c) 37 (d) 1
  32. 32.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10201 (b) 9801 (c) 10001 (d) 10211
  33. 33.If a + b = 11 and ab = 24, what is (a − b)²? (a) 121 (b) 73 (c) 5 (d) 25
  34. 34.What is the product of the roots of x² − 10x + 21 = 0? (a) 22 (b) 21 (c) 20 (d) 10
  35. 35.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4904 (b) 5194 (c) 4624 (d) 5184
  36. 36.What are the roots of x² − 11x + 30 = 0? (a) 2 and 15 (b) 5 and 6 (c) 1 and 10 (d) 1 and 30
  37. 37.If a + b = 8 and ab = 15, what is (a − b)²? (a) 64 (b) 2 (c) 4 (d) 34
  38. 38.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 1)(x + 35) (c) (x + 5)(x + 7) (d) (x + 1)(x + 11)
  39. 39.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 5625 (b) 4225 (c) 4235 (d) 4875
  40. 40.If a + b = 17 and ab = 72, what is (a − b)²? (a) 145 (b) 217 (c) 289 (d) 1

Answer key

  1. 8 and 9
  2. 15
  3. 170
  4. 16
  5. 5929
  6. (x + 2)(x + 6)
  7. 6 and 8
  8. 2 and 5
  9. 2704
  10. 125
  11. 9
  12. (x + 5)(x + 6)
  13. 49
  14. 16
  15. 10
  16. 81
  17. 113
  18. 12
  19. 576
  20. 3 and 8
  21. 37
  22. 25
  23. (x + 2)(x + 4)
  24. 1
  25. 441
  26. 8
  27. 3025
  28. (x + 8)(x + 9)
  29. 784
  30. 2304
  31. 1
  32. 10201
  33. 25
  34. 21
  35. 5184
  36. 5 and 6
  37. 4
  38. (x + 5)(x + 7)
  39. 4225
  40. 1

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