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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 225 · 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.If a + b = 19 and ab = 88, what is a² + b²? (a) 185 (b) 273 (c) 361 (d) 537
  2. 2.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 5184 (b) 4634 (c) 4624 (d) 4896
  3. 3.If a + b = 14 and ab = 45, what is (a − b)²? (a) 106 (b) 196 (c) 16 (d) 4
  4. 4.What are the roots of x² − 6x + 8 = 0? (a) 2 and 5 (b) 2 and 4 (c) 1 and 5 (d) 1 and 8
  5. 5.If a + b = 19 and ab = 90, what is a² + b²? (a) 181 (b) 271 (c) 541 (d) 361
  6. 6.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 5776 (b) 7066 (c) 6416 (d) 7056
  7. 7.What is the sum of the roots of x² − 10x + 21 = 0? (a) 11 (b) 10 (c) 21 (d) 9
  8. 8.What is the sum of the roots of x² − 9x + 14 = 0? (a) 8 (b) 10 (c) 9 (d) 14
  9. 9.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)
  10. 10.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 17 (c) 16 (d) 72
  11. 11.What is the sum of the roots of x² − 10x + 24 = 0? (a) 11 (b) 9 (c) 10 (d) 24
  12. 12.What is the sum of the roots of x² − 15x + 54 = 0? (a) 15 (b) 14 (c) 16 (d) 54
  13. 13.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2926 (b) 2116 (c) 2516 (d) 2916
  14. 14.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1596 (b) 1444 (c) 1454 (d) 1764
  15. 15.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1681 (b) 1599 (c) 1531 (d) 1521
  16. 16.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 18) (b) (x + 2)(x + 9) (c) (x + 1)(x + 10) (d) (x − 2)(x − 9)
  17. 17.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5051 (b) 4761 (c) 4901 (d) 5041
  18. 18.If a + b = 18 and ab = 80, what is a² + b²? (a) 244 (b) 484 (c) 324 (d) 164
  19. 19.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x − 6)(x − 7) (c) (x + 1)(x + 12) (d) (x + 6)(x + 7)
  20. 20.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 494 (b) 404 (c) 484 (d) 324
  21. 21.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x + 1)(x + 10) (c) (x − 3)(x − 8) (d) (x + 3)(x + 8)
  22. 22.If a + b = 18 and ab = 77, what is (a − b)²? (a) 170 (b) 16 (c) 4 (d) 324
  23. 23.If a + b = 12 and ab = 27, what is a² + b²? (a) 198 (b) 117 (c) 144 (d) 90
  24. 24.If a + b = 14 and ab = 48, what is (a − b)²? (a) 2 (b) 100 (c) 4 (d) 196
  25. 25.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6561 (b) 6571 (c) 6241 (d) 6401
  26. 26.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3374 (b) 3844 (c) 3596 (d) 3364
  27. 27.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3035 (b) 3575 (c) 3025 (d) 4225
  28. 28.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 416 (c) 586 (d) 576
  29. 29.What is the sum of the roots of x² − 7x + 10 = 0? (a) 6 (b) 10 (c) 8 (d) 7
  30. 30.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 289 (b) 409 (c) 539 (d) 529
  31. 31.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 7921 (b) 8101 (c) 8291 (d) 8281
  32. 32.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2509 (b) 2209 (c) 2809 (d) 2819
  33. 33.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 841 (b) 971 (c) 961 (d) 901
  34. 34.What is the sum of the roots of x² − 14x + 48 = 0? (a) 13 (b) 48 (c) 15 (d) 14
  35. 35.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 6)(x + 8) (c) (x + 1)(x + 48) (d) (x + 1)(x + 13)
  36. 36.What are the roots of x² − 15x + 56 = 0? (a) 7 and 8 (b) 2 and 28 (c) 1 and 56 (d) 1 and 14
  37. 37.If a + b = 5 and ab = 6, what is (a − b)²? (a) 1 (b) 13 (c) 25 (d) 19
  38. 38.If a + b = 8 and ab = 15, what is a² + b²? (a) 34 (b) 49 (c) 64 (d) 94
  39. 39.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x + 2)(x + 4) (c) (x + 1)(x + 8) (d) (x − 2)(x − 4)
  40. 40.If a + b = 12 and ab = 35, what is a² + b²? (a) 74 (b) 109 (c) 214 (d) 144

Answer key

  1. 185
  2. 4624
  3. 16
  4. 2 and 4
  5. 181
  6. 7056
  7. 10
  8. 9
  9. (x + 8)(x + 9)
  10. 17
  11. 10
  12. 15
  13. 2916
  14. 1444
  15. 1521
  16. (x + 2)(x + 9)
  17. 5041
  18. 164
  19. (x + 6)(x + 7)
  20. 484
  21. (x + 3)(x + 8)
  22. 16
  23. 90
  24. 4
  25. 6561
  26. 3364
  27. 3025
  28. 576
  29. 7
  30. 529
  31. 8281
  32. 2809
  33. 961
  34. 14
  35. (x + 6)(x + 8)
  36. 7 and 8
  37. 1
  38. 34
  39. (x + 2)(x + 4)
  40. 74

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