Question 26073
<pre>(2-4i)/(1+3i)
<font size = 4><b>
          3 - 4<i>i</i>
          ——————
          1 + 3<i>i</i>

The denominator is 1 + 3<i>i</i>.

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 -3<i>i</i>
   c. Putting these together
      1 - 3<i>i</i>
   d. Forming a fraction which equals one by placing this 
      conjugate over itself.
      1 - 3<i>i</i>
      ——————
      1 - 3<i>i</i>

2. Multiply original fraction by this fraction:

      3 - 4<i>i</i>   1 - 3<i>i</i>
      —————— × ——————
      1 + 3<i>i</i>   1 - 3<i>i</i>

3. Put parentheses around all numerators and denominators


      (3 - 4<i>i</i>)   (1 - 3<i>i</i>)
      ———————— × ————————
      (1 + 3<i>i</i>)   (1 - 3<i>i</i>)

4. Write as one fraction:

      (3 - 4<i>i</i>)(1 - 3<i>i</i>)
      ————————————————
      (1 + 3<i>i</i>)(1 - 3<i>i</i>)

5. Use FOIL to multiply out numerator and denominator

       3 - 9<i>i</i> - 4<i>i</i> + 12<i>i</i>²
      ————————————————————
       1 - 3<i>i</i> + 3<i>i</i> - 9<i>i</i>²

6. Combine terms in numerator and denominator

       3 - 13<i>i</i> + 12<i>i</i>²
      ———————————————
          1 - 9<i>i</i>²

7. Replace <i>i</i>² by its equivalent -1

       3 - 13<i>i</i> + 12(-1)
      ——————————————————
          1 - 9(-1)

8. Simplify

       3 - 13<i>i</i> - 12
      ——————————————————
          1 + 9

       -9 - 13<i>i</i>
      ——————————
          10

9.  Write as two fractions

       -9     13<i>i</i>
      ———— - —————
       10     10

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

       -9     13
      ———— - ———— <i>i</i>
       10     10 

Edwin
AnlytcPhil@aol.com</pre>