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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 503 · 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.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 2)(x + 4) (c) (x + 1)(x + 5) (d) (x + 1)(x + 8)
  2. 2.If a + b = 17 and ab = 66, what is (a − b)²? (a) 289 (b) 157 (c) 5 (d) 25
  3. 3.If a + b = 19 and ab = 88, what is (a − b)²? (a) 3 (b) 9 (c) 185 (d) 361
  4. 4.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 225 (c) 3 (d) 9
  5. 5.If a + b = 13 and ab = 36, what is (a − b)²? (a) 97 (b) 25 (c) 5 (d) 169
  6. 6.If a + b = 13 and ab = 22, what is (a − b)²? (a) 125 (b) 81 (c) 9 (d) 169
  7. 7.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x − 7)(x − 8) (c) (x + 1)(x + 14) (d) (x + 7)(x + 8)
  8. 8.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1601 (b) 1691 (c) 1521 (d) 1681
  9. 9.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6251 (b) 6241 (c) 6399 (d) 6561
  10. 10.What is the sum of the roots of x² − 13x + 42 = 0? (a) 42 (b) 14 (c) 13 (d) 12
  11. 11.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 6)(x + 9) (c) (x + 1)(x + 14) (d) (x − 6)(x − 9)
  12. 12.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 16 (c) 56 (d) 14
  13. 13.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)
  14. 14.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 2)(x + 7) (c) (x + 1)(x + 8) (d) (x + 1)(x + 14)
  15. 15.What are the roots of x² − 13x + 40 = 0? (a) 2 and 20 (b) 1 and 40 (c) 1 and 12 (d) 5 and 8
  16. 16.If a + b = 6 and ab = 5, what is a² + b²? (a) 31 (b) 36 (c) 26 (d) 46
  17. 17.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 8)(x + 9) (c) (x + 1)(x + 16) (d) (x + 1)(x + 72)
  18. 18.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 841 (b) 901 (c) 971 (d) 961
  19. 19.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)
  20. 20.What is the product of the roots of x² − 13x + 40 = 0? (a) 13 (b) 41 (c) 40 (d) 39
  21. 21.If a + b = 10 and ab = 21, what is (a − b)²? (a) 58 (b) 16 (c) 4 (d) 100
  22. 22.If a + b = 9 and ab = 18, what is a² + b²? (a) 63 (b) 45 (c) 117 (d) 81
  23. 23.If a + b = 8 and ab = 7, what is a² + b²? (a) 78 (b) 57 (c) 64 (d) 50
  24. 24.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 6)(x + 8) (c) (x + 1)(x + 48) (d) (x − 6)(x − 8)
  25. 25.What is the sum of the roots of x² − 12x + 32 = 0? (a) 32 (b) 12 (c) 11 (d) 13
  26. 26.What is the product of the roots of x² − 9x + 14 = 0? (a) 15 (b) 14 (c) 9 (d) 13
  27. 27.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x + 1)(x + 11) (c) (x + 5)(x + 7) (d) (x − 5)(x − 7)
  28. 28.If a + b = 15 and ab = 36, what is (a − b)²? (a) 153 (b) 81 (c) 9 (d) 225
  29. 29.If a + b = 5 and ab = 4, what is (a − b)²? (a) 3 (b) 17 (c) 9 (d) 25
  30. 30.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x + 3)(x + 4) (c) (x − 3)(x − 4) (d) (x + 1)(x + 12)
  31. 31.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9996 (b) 9614 (c) 10404 (d) 9604
  32. 32.If a + b = 14 and ab = 40, what is a² + b²? (a) 276 (b) 196 (c) 116 (d) 156
  33. 33.What is the sum of the roots of x² − 14x + 45 = 0? (a) 15 (b) 13 (c) 45 (d) 14
  34. 34.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4761 (b) 4771 (c) 4899 (d) 5041
  35. 35.If a + b = 8 and ab = 15, what is a² + b²? (a) 34 (b) 49 (c) 64 (d) 94
  36. 36.If a + b = 18 and ab = 77, what is (a − b)²? (a) 324 (b) 4 (c) 170 (d) 16
  37. 37.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 1024 (b) 794 (c) 896 (d) 784
  38. 38.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1616 (b) 1946 (c) 1296 (d) 1936
  39. 39.If a + b = 19 and ab = 84, what is a² + b²? (a) 361 (b) 193 (c) 277 (d) 529
  40. 40.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 5929 (b) 6889 (c) 6409 (d) 6899

Answer key

  1. (x + 2)(x + 4)
  2. 25
  3. 9
  4. 9
  5. 25
  6. 81
  7. (x + 7)(x + 8)
  8. 1681
  9. 6241
  10. 13
  11. (x + 6)(x + 9)
  12. 15
  13. (x + 6)(x + 7)
  14. (x + 2)(x + 7)
  15. 5 and 8
  16. 26
  17. (x + 8)(x + 9)
  18. 961
  19. (x + 3)(x + 8)
  20. 40
  21. 16
  22. 45
  23. 50
  24. (x + 6)(x + 8)
  25. 12
  26. 14
  27. (x + 5)(x + 7)
  28. 81
  29. 9
  30. (x + 3)(x + 4)
  31. 9604
  32. 116
  33. 14
  34. 4761
  35. 34
  36. 16
  37. 784
  38. 1936
  39. 193
  40. 6889

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