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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 443 · 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.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 729 (b) 909 (c) 1089 (d) 1099
  2. 2.If a + b = 5 and ab = 4, what is a² + b²? (a) 21 (b) 17 (c) 25 (d) 33
  3. 3.If a + b = 11 and ab = 28, what is a² + b²? (a) 65 (b) 93 (c) 121 (d) 177
  4. 4.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8474 (b) 8104 (c) 7744 (d) 8464
  5. 5.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3625 (b) 4235 (c) 3025 (d) 4225
  6. 6.If a + b = 15 and ab = 44, what is (a − b)²? (a) 49 (b) 7 (c) 225 (d) 137
  7. 7.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)
  8. 8.If a + b = 18 and ab = 77, what is (a − b)²? (a) 16 (b) 4 (c) 170 (d) 324
  9. 9.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x − 4)(x − 6) (c) (x + 1)(x + 9) (d) (x + 4)(x + 6)
  10. 10.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 10816 (b) 9226 (c) 9984 (d) 9216
  11. 11.If a + b = 8 and ab = 7, what is (a − b)²? (a) 50 (b) 6 (c) 36 (d) 64
  12. 12.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 6384 (b) 5786 (c) 7056 (d) 5776
  13. 13.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 334 (b) 324 (c) 396 (d) 484
  14. 14.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x − 4)(x − 5) (c) (x + 1)(x + 8) (d) (x + 1)(x + 20)
  15. 15.If a + b = 19 and ab = 88, what is (a − b)²? (a) 9 (b) 3 (c) 185 (d) 361
  16. 16.If a + b = 20 and ab = 96, what is a² + b²? (a) 400 (b) 208 (c) 304 (d) 592
  17. 17.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3969 (b) 3979 (c) 3249 (d) 3609
  18. 18.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 72 (c) 73 (d) 17
  19. 19.If a + b = 4 and ab = 3, what is a² + b²? (a) 22 (b) 16 (c) 13 (d) 10
  20. 20.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x − 2)(x − 6) (c) (x + 1)(x + 12) (d) (x + 2)(x + 6)
  21. 21.What are the roots of x² − 12x + 35 = 0? (a) 1 and 35 (b) 5 and 7 (c) 2 and 10 (d) 1 and 11
  22. 22.What is the sum of the roots of x² − 14x + 45 = 0? (a) 14 (b) 15 (c) 45 (d) 13
  23. 23.If a + b = 20 and ab = 96, what is (a − b)²? (a) 400 (b) 4 (c) 208 (d) 16
  24. 24.If a + b = 18 and ab = 72, what is (a − b)²? (a) 36 (b) 324 (c) 6 (d) 180
  25. 25.What is the product of the roots of x² − 10x + 24 = 0? (a) 24 (b) 25 (c) 23 (d) 10
  26. 26.If a + b = 12 and ab = 32, what is a² + b²? (a) 208 (b) 80 (c) 144 (d) 112
  27. 27.If a + b = 15 and ab = 44, what is a² + b²? (a) 137 (b) 181 (c) 225 (d) 313
  28. 28.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)
  29. 29.Which is the correct factorisation of x² + 9x + 14? (a) (x + 2)(x + 7) (b) (x + 1)(x + 8) (c) (x − 2)(x − 7) (d) (x + 1)(x + 14)
  30. 30.What is the sum of the roots of x² − 6x + 8 = 0? (a) 5 (b) 7 (c) 6 (d) 8
  31. 31.Which is the correct factorisation of x² + 11x + 18? (a) (x − 2)(x − 9) (b) (x + 2)(x + 9) (c) (x + 1)(x + 18) (d) (x + 1)(x + 10)
  32. 32.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)
  33. 33.What is the sum of the roots of x² − 12x + 32 = 0? (a) 11 (b) 13 (c) 32 (d) 12
  34. 34.If a + b = 10 and ab = 24, what is (a − b)²? (a) 4 (b) 52 (c) 100 (d) 2
  35. 35.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 12) (c) (x + 3)(x + 4) (d) (x + 1)(x + 6)
  36. 36.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x + 4)(x + 8) (c) (x + 1)(x + 11) (d) (x − 4)(x − 8)
  37. 37.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2601 (b) 2499 (c) 2401 (d) 2411
  38. 38.If a + b = 13 and ab = 30, what is (a − b)²? (a) 109 (b) 7 (c) 49 (d) 169
  39. 39.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1225 (b) 1235 (c) 925 (d) 625
  40. 40.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8659 (b) 7569 (c) 8649 (d) 8109

Answer key

  1. 1089
  2. 17
  3. 65
  4. 8464
  5. 4225
  6. 49
  7. (x + 8)(x + 9)
  8. 16
  9. (x + 4)(x + 6)
  10. 9216
  11. 36
  12. 5776
  13. 324
  14. (x + 4)(x + 5)
  15. 9
  16. 208
  17. 3969
  18. 72
  19. 10
  20. (x + 2)(x + 6)
  21. 5 and 7
  22. 14
  23. 16
  24. 36
  25. 24
  26. 80
  27. 137
  28. (x + 6)(x + 9)
  29. (x + 2)(x + 7)
  30. 6
  31. (x + 2)(x + 9)
  32. (x + 7)(x + 8)
  33. 12
  34. 4
  35. (x + 3)(x + 4)
  36. (x + 4)(x + 8)
  37. 2401
  38. 49
  39. 1225
  40. 8649

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