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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 534 · 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 (a − b)²? (a) 361 (b) 185 (c) 9 (d) 3
  2. 2.If a + b = 7 and ab = 6, what is a² + b²? (a) 43 (b) 37 (c) 61 (d) 49
  3. 3.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 7)(x + 8) (c) (x − 7)(x − 8) (d) (x + 1)(x + 56)
  4. 4.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x − 3)(x − 8) (c) (x + 3)(x + 8) (d) (x + 1)(x + 10)
  5. 5.What are the roots of x² − 10x + 16 = 0? (a) 3 and 7 (b) 1 and 16 (c) 2 and 8 (d) 1 and 9
  6. 6.If a + b = 9 and ab = 20, what is (a − b)²? (a) 61 (b) 81 (c) 1 (d) 41
  7. 7.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2611 (b) 2401 (c) 2501 (d) 2601
  8. 8.If a + b = 14 and ab = 48, what is (a − b)²? (a) 4 (b) 196 (c) 100 (d) 2
  9. 9.If a + b = 17 and ab = 66, what is (a − b)²? (a) 289 (b) 5 (c) 157 (d) 25
  10. 10.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 896 (c) 784 (d) 1024
  11. 11.If a + b = 14 and ab = 40, what is a² + b²? (a) 196 (b) 156 (c) 116 (d) 276
  12. 12.If a + b = 16 and ab = 63, what is (a − b)²? (a) 2 (b) 130 (c) 4 (d) 256
  13. 13.What is the product of the roots of x² − 13x + 36 = 0? (a) 35 (b) 36 (c) 37 (d) 13
  14. 14.If a + b = 7 and ab = 10, what is a² + b²? (a) 69 (b) 29 (c) 39 (d) 49
  15. 15.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3374 (b) 3844 (c) 3364 (d) 3596
  16. 16.What is the product of the roots of x² − 5x + 6 = 0? (a) 6 (b) 7 (c) 8 (d) 5
  17. 17.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x + 1)(x + 12) (c) (x − 6)(x − 7) (d) (x + 1)(x + 42)
  18. 18.If a + b = 17 and ab = 72, what is a² + b²? (a) 145 (b) 433 (c) 217 (d) 289
  19. 19.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1296 (b) 1584 (c) 1306 (d) 1936
  20. 20.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 676 (b) 884 (c) 1156 (d) 686
  21. 21.If a + b = 18 and ab = 77, what is a² + b²? (a) 170 (b) 478 (c) 324 (d) 247
  22. 22.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 625 (b) 925 (c) 1235 (d) 1225
  23. 23.What is the sum of the roots of x² − 10x + 24 = 0? (a) 11 (b) 24 (c) 10 (d) 9
  24. 24.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8464 (b) 7744 (c) 8474 (d) 8104
  25. 25.What is the sum of the roots of x² − 11x + 24 = 0? (a) 11 (b) 24 (c) 12 (d) 10
  26. 26.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4489 (b) 4499 (c) 5329 (d) 4891
  27. 27.If a + b = 19 and ab = 84, what is (a − b)²? (a) 25 (b) 193 (c) 5 (d) 361
  28. 28.Which is the correct factorisation of x² + 7x + 10? (a) (x + 2)(x + 5) (b) (x + 1)(x + 6) (c) (x − 2)(x − 5) (d) (x + 1)(x + 10)
  29. 29.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 451 (b) 401 (c) 441 (d) 361
  30. 30.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
  31. 31.If a + b = 15 and ab = 50, what is (a − b)²? (a) 225 (b) 125 (c) 5 (d) 25
  32. 32.If a + b = 6 and ab = 5, what is a² + b²? (a) 46 (b) 31 (c) 36 (d) 26
  33. 33.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10826 (b) 9216 (c) 10816 (d) 10016
  34. 34.If a + b = 7 and ab = 12, what is (a − b)²? (a) 49 (b) 25 (c) 1 (d) 37
  35. 35.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1296 (b) 1616 (c) 1936 (d) 1946
  36. 36.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x + 2)(x + 7) (c) (x + 1)(x + 8) (d) (x − 2)(x − 7)
  37. 37.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 17 (c) 72 (d) 18
  38. 38.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 1)(x + 63) (c) (x + 7)(x + 9) (d) (x − 7)(x − 9)
  39. 39.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 14 (c) 56 (d) 15
  40. 40.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 7225 (b) 9035 (c) 8125 (d) 9025

Answer key

  1. 9
  2. 37
  3. (x + 7)(x + 8)
  4. (x + 3)(x + 8)
  5. 2 and 8
  6. 1
  7. 2601
  8. 4
  9. 25
  10. 784
  11. 116
  12. 4
  13. 36
  14. 29
  15. 3364
  16. 6
  17. (x + 6)(x + 7)
  18. 145
  19. 1296
  20. 676
  21. 170
  22. 1225
  23. 10
  24. 8464
  25. 11
  26. 4489
  27. 25
  28. (x + 2)(x + 5)
  29. 441
  30. (x + 8)(x + 9)
  31. 25
  32. 26
  33. 10816
  34. 1
  35. 1936
  36. (x + 2)(x + 7)
  37. 17
  38. (x + 7)(x + 9)
  39. 15
  40. 9025

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