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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 726 · 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 = 9 and ab = 8, what is a² + b²? (a) 81 (b) 73 (c) 65 (d) 97
  2. 2.If a + b = 18 and ab = 77, what is a² + b²? (a) 170 (b) 478 (c) 247 (d) 324
  3. 3.What are the roots of x² − 12x + 35 = 0? (a) 1 and 35 (b) 2 and 10 (c) 1 and 11 (d) 5 and 7
  4. 4.If a + b = 11 and ab = 28, what is a² + b²? (a) 121 (b) 177 (c) 93 (d) 65
  5. 5.Which is the correct factorisation of x² + 13x + 40? (a) (x + 5)(x + 8) (b) (x − 5)(x − 8) (c) (x + 1)(x + 12) (d) (x + 1)(x + 40)
  6. 6.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)
  7. 7.What is the sum of the roots of x² − 9x + 14 = 0? (a) 14 (b) 9 (c) 10 (d) 8
  8. 8.If a + b = 20 and ab = 96, what is a² + b²? (a) 208 (b) 592 (c) 304 (d) 400
  9. 9.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)
  10. 10.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2025 (b) 2525 (c) 3035 (d) 3025
  11. 11.If a + b = 11 and ab = 28, what is (a − b)²? (a) 3 (b) 121 (c) 65 (d) 9
  12. 12.Which is the correct factorisation of x² + 10x + 16? (a) (x + 2)(x + 8) (b) (x + 1)(x + 9) (c) (x − 2)(x − 8) (d) (x + 1)(x + 16)
  13. 13.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1521 (b) 1601 (c) 1681 (d) 1691
  14. 14.If a + b = 13 and ab = 22, what is (a − b)²? (a) 169 (b) 125 (c) 81 (d) 9
  15. 15.What is the product of the roots of x² − 16x + 63 = 0? (a) 16 (b) 64 (c) 62 (d) 63
  16. 16.What is the product of the roots of x² − 6x + 8 = 0? (a) 8 (b) 9 (c) 7 (d) 6
  17. 17.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 1089 (b) 891 (c) 739 (d) 729
  18. 18.What is the product of the roots of x² − 13x + 42 = 0? (a) 43 (b) 42 (c) 41 (d) 13
  19. 19.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 494 (b) 484 (c) 324 (d) 404
  20. 20.If a + b = 7 and ab = 10, what is (a − b)²? (a) 29 (b) 9 (c) 3 (d) 49
  21. 21.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 1)(x + 30) (c) (x + 5)(x + 6) (d) (x − 5)(x − 6)
  22. 22.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3604 (b) 3854 (c) 3844 (d) 3364
  23. 23.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 361 (c) 541 (d) 181
  24. 24.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 56 (c) 15 (d) 14
  25. 25.If a + b = 15 and ab = 36, what is a² + b²? (a) 297 (b) 225 (c) 189 (d) 153
  26. 26.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 400 (c) 4 (d) 202
  27. 27.Which is the correct factorisation of x² + 12x + 32? (a) (x + 4)(x + 8) (b) (x + 1)(x + 32) (c) (x − 4)(x − 8) (d) (x + 1)(x + 11)
  28. 28.If a + b = 12 and ab = 35, what is a² + b²? (a) 109 (b) 144 (c) 74 (d) 214
  29. 29.If a + b = 6 and ab = 8, what is (a − b)²? (a) 2 (b) 36 (c) 4 (d) 20
  30. 30.What is the sum of the roots of x² − 15x + 54 = 0? (a) 15 (b) 54 (c) 16 (d) 14
  31. 31.If a + b = 18 and ab = 72, what is (a − b)²? (a) 180 (b) 324 (c) 36 (d) 6
  32. 32.If a + b = 18 and ab = 72, what is a² + b²? (a) 324 (b) 468 (c) 252 (d) 180
  33. 33.What is the product of the roots of x² − 11x + 28 = 0? (a) 27 (b) 11 (c) 29 (d) 28
  34. 34.What are the roots of x² − 13x + 42 = 0? (a) 1 and 12 (b) 2 and 21 (c) 6 and 7 (d) 1 and 42
  35. 35.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x + 1)(x + 13) (c) (x + 1)(x + 45) (d) (x − 5)(x − 9)
  36. 36.If a + b = 17 and ab = 60, what is a² + b²? (a) 409 (b) 289 (c) 169 (d) 229
  37. 37.What are the roots of x² − 8x + 12 = 0? (a) 1 and 12 (b) 1 and 7 (c) 3 and 4 (d) 2 and 6
  38. 38.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 6)(x + 8) (c) (x + 1)(x + 48) (d) (x + 1)(x + 13)
  39. 39.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1609 (b) 1849 (c) 1859 (d) 1369
  40. 40.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 841 (b) 961 (c) 899 (d) 851

Answer key

  1. 65
  2. 170
  3. 5 and 7
  4. 65
  5. (x + 5)(x + 8)
  6. (x + 3)(x + 5)
  7. 9
  8. 208
  9. (x + 8)(x + 9)
  10. 3025
  11. 9
  12. (x + 2)(x + 8)
  13. 1681
  14. 81
  15. 63
  16. 8
  17. 729
  18. 42
  19. 484
  20. 9
  21. (x + 5)(x + 6)
  22. 3844
  23. 181
  24. 15
  25. 153
  26. 4
  27. (x + 4)(x + 8)
  28. 74
  29. 4
  30. 15
  31. 36
  32. 180
  33. 28
  34. 6 and 7
  35. (x + 5)(x + 9)
  36. 169
  37. 2 and 6
  38. (x + 6)(x + 8)
  39. 1849
  40. 841

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