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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 689 · 40 questions
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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 59²? (a) 3599 (b) 3481 (c) 3491 (d) 3721
  2. 2.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 400 (c) 301 (d) 202
  3. 3.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 625 (b) 1225 (c) 635 (d) 875
  4. 4.What is the product of the roots of x² − 12x + 27 = 0? (a) 26 (b) 27 (c) 28 (d) 12
  5. 5.What is the sum of the roots of x² − 15x + 54 = 0? (a) 16 (b) 54 (c) 14 (d) 15
  6. 6.If a + b = 18 and ab = 80, what is a² + b²? (a) 164 (b) 484 (c) 324 (d) 244
  7. 7.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x + 3)(x + 7) (c) (x + 1)(x + 9) (d) (x − 3)(x − 7)
  8. 8.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 16 (c) 18 (d) 72
  9. 9.If a + b = 11 and ab = 10, what is a² + b²? (a) 111 (b) 121 (c) 101 (d) 141
  10. 10.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7754 (b) 7744 (c) 8464 (d) 8096
  11. 11.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 425 (b) 625 (c) 225 (d) 635
  12. 12.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)
  13. 13.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)
  14. 14.If a + b = 13 and ab = 42, what is (a − b)²? (a) 169 (b) 1 (c) 85 (d) 127
  15. 15.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 1)(x + 11) (c) (x + 1)(x + 27) (d) (x + 3)(x + 9)
  16. 16.What are the roots of x² − 9x + 14 = 0? (a) 1 and 14 (b) 1 and 8 (c) 2 and 7 (d) 3 and 6
  17. 17.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 901 (b) 961 (c) 971 (d) 841
  18. 18.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9991 (b) 10609 (c) 9419 (d) 9409
  19. 19.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1764 (b) 1444 (c) 1604 (d) 1774
  20. 20.If a + b = 19 and ab = 90, what is (a − b)²? (a) 1 (b) 361 (c) 181 (d) 271
  21. 21.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)
  22. 22.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 8084 (b) 7406 (c) 8836 (d) 7396
  23. 23.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 1)(x + 13) (c) (x + 5)(x + 9) (d) (x + 1)(x + 45)
  24. 24.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 399 (b) 441 (c) 371 (d) 361
  25. 25.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x − 3)(x − 5) (c) (x + 1)(x + 15) (d) (x + 3)(x + 5)
  26. 26.What is the sum of the roots of x² − 12x + 35 = 0? (a) 13 (b) 12 (c) 11 (d) 35
  27. 27.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8464 (b) 7744 (c) 8104 (d) 8474
  28. 28.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 17 (c) 73 (d) 72
  29. 29.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 14 (c) 16 (d) 15
  30. 30.What is the sum of the roots of x² − 6x + 8 = 0? (a) 6 (b) 8 (c) 5 (d) 7
  31. 31.If a + b = 6 and ab = 5, what is a² + b²? (a) 31 (b) 36 (c) 26 (d) 46
  32. 32.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x + 1)(x + 14) (c) (x + 1)(x + 54) (d) (x − 6)(x − 9)
  33. 33.If a + b = 15 and ab = 54, what is (a − b)²? (a) 117 (b) 9 (c) 225 (d) 3
  34. 34.If a + b = 10 and ab = 16, what is (a − b)²? (a) 68 (b) 100 (c) 6 (d) 36
  35. 35.If a + b = 18 and ab = 77, what is a² + b²? (a) 247 (b) 170 (c) 478 (d) 324
  36. 36.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 10025 (b) 11035 (c) 11025 (d) 9025
  37. 37.If a + b = 5 and ab = 4, what is (a − b)²? (a) 17 (b) 25 (c) 3 (d) 9
  38. 38.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 63 (c) 16 (d) 15
  39. 39.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 5476 (b) 4366 (c) 4356 (d) 4884
  40. 40.If a + b = 16 and ab = 48, what is (a − b)²? (a) 256 (b) 160 (c) 8 (d) 64

Answer key

  1. 3481
  2. 202
  3. 625
  4. 27
  5. 15
  6. 164
  7. (x + 3)(x + 7)
  8. 17
  9. 101
  10. 7744
  11. 625
  12. (x + 8)(x + 9)
  13. (x + 4)(x + 6)
  14. 1
  15. (x + 3)(x + 9)
  16. 2 and 7
  17. 961
  18. 9409
  19. 1764
  20. 1
  21. (x + 7)(x + 9)
  22. 7396
  23. (x + 5)(x + 9)
  24. 361
  25. (x + 3)(x + 5)
  26. 12
  27. 8464
  28. 72
  29. 15
  30. 6
  31. 26
  32. (x + 6)(x + 9)
  33. 9
  34. 36
  35. 170
  36. 11025
  37. 9
  38. 16
  39. 4356
  40. 64

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