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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 27 · 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.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6409 (b) 6899 (c) 6889 (d) 5929
  2. 2.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
  3. 3.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 72 (c) 71 (d) 73
  4. 4.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 1)(x + 20) (c) (x + 4)(x + 5) (d) (x + 1)(x + 8)
  5. 5.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4634 (b) 4896 (c) 5184 (d) 4624
  6. 6.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 441 (b) 371 (c) 361 (d) 399
  7. 7.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 4909 (b) 4489 (c) 5339 (d) 5329
  8. 8.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x + 5)(x + 7) (c) (x + 1)(x + 11) (d) (x − 5)(x − 7)
  9. 9.If a + b = 18 and ab = 77, what is a² + b²? (a) 247 (b) 170 (c) 324 (d) 478
  10. 10.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 7056 (b) 5776 (c) 6384 (d) 5786
  11. 11.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3844 (b) 3364 (c) 3374 (d) 3596
  12. 12.If a + b = 15 and ab = 50, what is a² + b²? (a) 175 (b) 325 (c) 225 (d) 125
  13. 13.What is the product of the roots of x² − 14x + 48 = 0? (a) 48 (b) 14 (c) 49 (d) 47
  14. 14.If a + b = 19 and ab = 88, what is a² + b²? (a) 537 (b) 361 (c) 185 (d) 273
  15. 15.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1774 (b) 1444 (c) 1604 (d) 1764
  16. 16.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 16 (c) 63 (d) 17
  17. 17.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x + 1)(x + 13) (c) (x − 6)(x − 8) (d) (x + 6)(x + 8)
  18. 18.If a + b = 5 and ab = 6, what is (a − b)²? (a) 19 (b) 25 (c) 1 (d) 13
  19. 19.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 5635 (b) 4925 (c) 4225 (d) 5625
  20. 20.If a + b = 14 and ab = 40, what is (a − b)²? (a) 116 (b) 196 (c) 6 (d) 36
  21. 21.What are the roots of x² − 11x + 18 = 0? (a) 1 and 10 (b) 1 and 18 (c) 3 and 6 (d) 2 and 9
  22. 22.If a + b = 18 and ab = 80, what is a² + b²? (a) 244 (b) 324 (c) 484 (d) 164
  23. 23.If a + b = 12 and ab = 20, what is (a − b)²? (a) 64 (b) 144 (c) 104 (d) 8
  24. 24.What is the product of the roots of x² − 9x + 20 = 0? (a) 9 (b) 21 (c) 19 (d) 20
  25. 25.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x − 3)(x − 5) (c) (x + 3)(x + 5) (d) (x + 1)(x + 7)
  26. 26.If a + b = 19 and ab = 84, what is a² + b²? (a) 361 (b) 277 (c) 529 (d) 193
  27. 27.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 901 (b) 841 (c) 971 (d) 961
  28. 28.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8104 (b) 7744 (c) 8474 (d) 8464
  29. 29.Which is the correct factorisation of x² + 11x + 18? (a) (x + 2)(x + 9) (b) (x + 1)(x + 18) (c) (x − 2)(x − 9) (d) (x + 1)(x + 10)
  30. 30.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6094 (b) 6724 (c) 6396 (d) 6084
  31. 31.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6561 (b) 6571 (c) 6241 (d) 6401
  32. 32.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 5)(x + 6) (c) (x + 1)(x + 30) (d) (x + 1)(x + 10)
  33. 33.If a + b = 11 and ab = 28, what is a² + b²? (a) 65 (b) 177 (c) 93 (d) 121
  34. 34.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 72 (c) 17 (d) 18
  35. 35.If a + b = 16 and ab = 63, what is a² + b²? (a) 382 (b) 256 (c) 130 (d) 193
  36. 36.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 333 (c) 225 (d) 657
  37. 37.What are the roots of x² − 15x + 56 = 0? (a) 2 and 28 (b) 1 and 14 (c) 1 and 56 (d) 7 and 8
  38. 38.If a + b = 13 and ab = 12, what is a² + b²? (a) 145 (b) 157 (c) 169 (d) 193
  39. 39.What is the product of the roots of x² − 11x + 30 = 0? (a) 31 (b) 11 (c) 29 (d) 30
  40. 40.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9419 (b) 10609 (c) 9409 (d) 9991

Answer key

  1. 6889
  2. (x + 8)(x + 9)
  3. 72
  4. (x + 4)(x + 5)
  5. 4624
  6. 361
  7. 5329
  8. (x + 5)(x + 7)
  9. 170
  10. 5776
  11. 3364
  12. 125
  13. 48
  14. 185
  15. 1764
  16. 16
  17. (x + 6)(x + 8)
  18. 1
  19. 5625
  20. 36
  21. 2 and 9
  22. 164
  23. 64
  24. 20
  25. (x + 3)(x + 5)
  26. 193
  27. 961
  28. 8464
  29. (x + 2)(x + 9)
  30. 6084
  31. 6561
  32. (x + 5)(x + 6)
  33. 65
  34. 17
  35. 130
  36. 225
  37. 7 and 8
  38. 145
  39. 30
  40. 9409

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