SOLUTION: simplify (2-4i)/(1+3i)

(2-4i)/(1+3i)

          3 - 4i
          ——————
          1 + 3i

The denominator is 1 + 3i.

I. Form its conjugate by: 
   a. Keeping the first term as it is:
      1
   b. Changing the sign of the second term
      change +3i to -3i
   c. Putting these together
      1 - 3i
   d. Forming a fraction which equals one by placing this 
      conjugate over itself.
      1 - 3i
      ——————
      1 - 3i

2. Multiply original fraction by this fraction:

      3 - 4i   1 - 3i
      —————— × ——————
      1 + 3i   1 - 3i

3. Put parentheses around all numerators and denominators


      (3 - 4i)   (1 - 3i)
      ———————— × ————————
      (1 + 3i)   (1 - 3i)

4. Write as one fraction:

      (3 - 4i)(1 - 3i)
      ————————————————
      (1 + 3i)(1 - 3i)

5. Use FOIL to multiply out numerator and denominator

       3 - 9i - 4i + 12i²
      ————————————————————
       1 - 3i + 3i - 9i²

6. Combine terms in numerator and denominator

       3 - 13i + 12i²
      ———————————————
          1 - 9i²

7. Replace i² by its equivalent -1

       3 - 13i + 12(-1)
      ——————————————————
          1 - 9(-1)

8. Simplify

       3 - 13i - 12
      ——————————————————
          1 + 9

       -9 - 13i
      ——————————
          10

9.  Write as two fractions

       -9     13i
      ———— - —————
       10     10

10. Write i as multiplied by the fraction 13/10

       -9     13
      ———— - ———— i
       10     10 

Edwin
AnlytcPhil@aol.com