Answer:
The answers are:
[tex]a=\frac{1}{7-4i} \\b=\frac{1}{13}[/tex]
Step-by-step explanation:
In order to determine the values of both a and b variables, we have to know a property of the imaginary numbers. The imaginary unit, [tex]i[/tex] , has the following rule:
[tex]i=\sqrt{-1}\\i^2=i*i=-1[/tex]
Then, we add both fraction inside of the parenthesis:
[tex]\frac{3+2i}{2-3i} +\frac{5-i}{2+3i}\\\frac{(3+2i)*(2+3i)+(5-i)*(2-3i)}{(2-3i)*(2+3i)}\\\frac{6+9i+4i+6i^2+10-15i-2i+3i^2}{4+6i-6i-9i^2}\\\frac{16-4i+9i^2}{4-9i^2}\\\frac{16-4i+9*(-1)}{4-9*(-1)}\\\frac{7-4i}{13}[/tex]
Then, if we multiply the numerator by a, we have to get 1. Then if we multiply by b the denominator, we have to get 1. So:
[tex]a=\frac{1}{7-4i} \\b=\frac{1}{13}[/tex]
