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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 829 · 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 = 18 and ab = 72, what is (a − b)²? (a) 36 (b) 324 (c) 180 (d) 6
  2. 2.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 16 (c) 17 (d) 72
  3. 3.If a + b = 18 and ab = 77, what is a² + b²? (a) 170 (b) 324 (c) 478 (d) 247
  4. 4.If a + b = 10 and ab = 21, what is (a − b)²? (a) 16 (b) 4 (c) 100 (d) 58
  5. 5.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 1225 (b) 625 (c) 635 (d) 875
  6. 6.If a + b = 16 and ab = 48, what is a² + b²? (a) 208 (b) 160 (c) 352 (d) 256
  7. 7.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4366 (b) 4356 (c) 5476 (d) 4884
  8. 8.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1369 (b) 1609 (c) 1849 (d) 1859
  9. 9.What is the product of the roots of x² − 14x + 45 = 0? (a) 46 (b) 44 (c) 45 (d) 14
  10. 10.If a + b = 13 and ab = 36, what is (a − b)²? (a) 97 (b) 5 (c) 169 (d) 25
  11. 11.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4225 (b) 5625 (c) 5635 (d) 4925
  12. 12.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x − 2)(x − 6) (c) (x + 1)(x + 7) (d) (x + 2)(x + 6)
  13. 13.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 333 (c) 225 (d) 657
  14. 14.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 27) (b) (x − 3)(x − 9) (c) (x + 3)(x + 9) (d) (x + 1)(x + 11)
  15. 15.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 6)(x + 8) (c) (x + 1)(x + 13) (d) (x + 1)(x + 48)
  16. 16.If a + b = 12 and ab = 20, what is a² + b²? (a) 144 (b) 124 (c) 184 (d) 104
  17. 17.If a + b = 13 and ab = 22, what is (a − b)²? (a) 81 (b) 9 (c) 125 (d) 169
  18. 18.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 13 (c) 15 (d) 14
  19. 19.What are the roots of x² − 15x + 56 = 0? (a) 7 and 8 (b) 1 and 14 (c) 2 and 28 (d) 1 and 56
  20. 20.If a + b = 16 and ab = 63, what is (a − b)²? (a) 256 (b) 130 (c) 4 (d) 2
  21. 21.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 5329 (b) 5339 (c) 4489 (d) 4909
  22. 22.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 57 (c) 55 (d) 56
  23. 23.What is the sum of the roots of x² − 14x + 45 = 0? (a) 15 (b) 45 (c) 14 (d) 13
  24. 24.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 10201 (b) 9811 (c) 9801 (d) 9999
  25. 25.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3721 (b) 3731 (c) 3601 (d) 3481
  26. 26.What is the product of the roots of x² − 12x + 35 = 0? (a) 36 (b) 12 (c) 35 (d) 34
  27. 27.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 266 (b) 256 (c) 576 (d) 384
  28. 28.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 72)
  29. 29.If a + b = 9 and ab = 18, what is a² + b²? (a) 63 (b) 117 (c) 81 (d) 45
  30. 30.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 371 (b) 441 (c) 399 (d) 361
  31. 31.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x − 5)(x − 9) (c) (x + 1)(x + 45) (d) (x + 5)(x + 9)
  32. 32.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1691 (b) 1681 (c) 1601 (d) 1521
  33. 33.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 14) (c) (x + 7)(x + 8) (d) (x + 1)(x + 56)
  34. 34.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2504 (b) 2714 (c) 2704 (d) 2304
  35. 35.What is the sum of the roots of x² − 13x + 42 = 0? (a) 14 (b) 13 (c) 12 (d) 42
  36. 36.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2525 (b) 3025 (c) 3035 (d) 2025
  37. 37.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2491 (b) 2809 (c) 2219 (d) 2209
  38. 38.If a + b = 15 and ab = 56, what is (a − b)²? (a) 1 (b) 169 (c) 113 (d) 225
  39. 39.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 299 (b) 391 (c) 529 (d) 289
  40. 40.If a + b = 5 and ab = 6, what is (a − b)²? (a) 19 (b) 25 (c) 13 (d) 1

Answer key

  1. 36
  2. 17
  3. 170
  4. 16
  5. 625
  6. 160
  7. 4356
  8. 1849
  9. 45
  10. 25
  11. 5625
  12. (x + 2)(x + 6)
  13. 225
  14. (x + 3)(x + 9)
  15. (x + 6)(x + 8)
  16. 104
  17. 81
  18. 14
  19. 7 and 8
  20. 4
  21. 5329
  22. 56
  23. 14
  24. 9801
  25. 3721
  26. 35
  27. 256
  28. (x + 8)(x + 9)
  29. 45
  30. 361
  31. (x + 5)(x + 9)
  32. 1681
  33. (x + 7)(x + 8)
  34. 2704
  35. 13
  36. 3025
  37. 2209
  38. 1
  39. 289
  40. 1

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