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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 360 · 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 = 18, what is (a − b)²? (a) 81 (b) 3 (c) 9 (d) 45
  2. 2.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x − 6)(x − 9) (c) (x + 1)(x + 54) (d) (x + 6)(x + 9)
  3. 3.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x − 5)(x − 6) (c) (x + 1)(x + 30) (d) (x + 5)(x + 6)
  4. 4.If a + b = 15 and ab = 50, what is (a − b)²? (a) 25 (b) 225 (c) 125 (d) 5
  5. 5.What is the sum of the roots of x² − 10x + 16 = 0? (a) 10 (b) 11 (c) 16 (d) 9
  6. 6.If a + b = 12 and ab = 35, what is (a − b)²? (a) 144 (b) 4 (c) 74 (d) 2
  7. 7.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 73 (c) 71 (d) 72
  8. 8.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2401 (b) 2601 (c) 2499 (d) 2411
  9. 9.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1166 (b) 676 (c) 916 (d) 1156
  10. 10.If a + b = 7 and ab = 10, what is (a − b)²? (a) 49 (b) 9 (c) 3 (d) 29
  11. 11.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 12) (c) (x + 1)(x + 40) (d) (x + 5)(x + 8)
  12. 12.If a + b = 11 and ab = 10, what is a² + b²? (a) 141 (b) 111 (c) 101 (d) 121
  13. 13.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 11025 (b) 9035 (c) 9975 (d) 9025
  14. 14.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2809 (b) 2209 (c) 2219 (d) 2491
  15. 15.If a + b = 21 and ab = 108, what is a² + b²? (a) 225 (b) 333 (c) 657 (d) 441
  16. 16.If a + b = 17 and ab = 72, what is (a − b)²? (a) 217 (b) 289 (c) 145 (d) 1
  17. 17.If a + b = 16 and ab = 55, what is (a − b)²? (a) 256 (b) 146 (c) 36 (d) 6
  18. 18.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 1)(x + 27) (c) (x + 1)(x + 11) (d) (x + 3)(x + 9)
  19. 19.Which is the correct factorisation of x² + 8x + 12? (a) (x − 2)(x − 6) (b) (x + 2)(x + 6) (c) (x + 1)(x + 12) (d) (x + 1)(x + 7)
  20. 20.Which is the correct factorisation of x² + 12x + 35? (a) (x + 5)(x + 7) (b) (x + 1)(x + 11) (c) (x − 5)(x − 7) (d) (x + 1)(x + 35)
  21. 21.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
  22. 22.If a + b = 8 and ab = 7, what is (a − b)²? (a) 64 (b) 6 (c) 50 (d) 36
  23. 23.If a + b = 9 and ab = 20, what is a² + b²? (a) 81 (b) 61 (c) 121 (d) 41
  24. 24.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)
  25. 25.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 7)(x + 9) (c) (x + 1)(x + 15) (d) (x + 1)(x + 63)
  26. 26.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3601 (b) 3731 (c) 3481 (d) 3721
  27. 27.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7235 (b) 5625 (c) 7225 (d) 6425
  28. 28.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 334 (b) 324 (c) 484 (d) 396
  29. 29.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 145 (c) 217 (d) 289
  30. 30.If a + b = 17 and ab = 66, what is a² + b²? (a) 223 (b) 421 (c) 157 (d) 289
  31. 31.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8846 (b) 8116 (c) 8836 (d) 7396
  32. 32.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)
  33. 33.If a + b = 10 and ab = 24, what is (a − b)²? (a) 4 (b) 2 (c) 100 (d) 52
  34. 34.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 416 (b) 576 (c) 256 (d) 586
  35. 35.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x − 8)(x − 9) (c) (x + 8)(x + 9) (d) (x + 1)(x + 16)
  36. 36.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 13) (c) (x + 1)(x + 48) (d) (x + 6)(x + 8)
  37. 37.What are the roots of x² − 9x + 14 = 0? (a) 1 and 8 (b) 2 and 7 (c) 1 and 14 (d) 3 and 6
  38. 38.If a + b = 16 and ab = 60, what is (a − b)²? (a) 16 (b) 4 (c) 136 (d) 256
  39. 39.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3969 (b) 3979 (c) 3249 (d) 3609
  40. 40.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 16 (c) 17 (d) 72

Answer key

  1. 9
  2. (x + 6)(x + 9)
  3. (x + 5)(x + 6)
  4. 25
  5. 10
  6. 4
  7. 72
  8. 2401
  9. 1156
  10. 9
  11. (x + 5)(x + 8)
  12. 101
  13. 9025
  14. 2209
  15. 225
  16. 1
  17. 36
  18. (x + 3)(x + 9)
  19. (x + 2)(x + 6)
  20. (x + 5)(x + 7)
  21. 6 and 7
  22. 36
  23. 41
  24. (x + 6)(x + 7)
  25. (x + 7)(x + 9)
  26. 3721
  27. 7225
  28. 324
  29. 145
  30. 157
  31. 8836
  32. (x + 7)(x + 8)
  33. 4
  34. 576
  35. (x + 8)(x + 9)
  36. (x + 6)(x + 8)
  37. 2 and 7
  38. 16
  39. 3969
  40. 17

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