To solve the equation [tex]\(\left(\frac{3+2i}{2-3i}+\frac{5-i}{2+3i}\right) \times \frac{a}{b}=1\)[/tex], we need to determine the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] that satisfy the equation.
First, we handle each fraction separately:
1. Simplify [tex]\(\frac{3+2i}{2-3i}\)[/tex].
2. Simplify [tex]\(\frac{5-i}{2+3i}\)[/tex].
Next, add these simplified fractions together to form a single combined fraction.
We then set this combined fraction multiplied by [tex]\(\frac{a}{b}\)[/tex] equal to 1.
Thus, we solve:
[tex]\[
\left(\frac{3+2i}{2-3i} + \frac{5-i}{2+3i}\right) \times \frac{a}{b} = 1
\][/tex]
From the problem, we know that the combined fraction evaluates to [tex]\(\frac{a}{b}=1\)[/tex].
Thus, if this combined fraction multiplied by [tex]\(\frac{a}{b}\)[/tex] equals 1, we know:
[tex]\[
\frac{a}{b} = \frac{2}{1}
\][/tex]
Therefore, the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are:
[tex]\[
a = 2
\][/tex]
[tex]\[
b = 1
\][/tex]
Hence, the completion of the statement would be:
If [tex]\(\left(\frac{3+2i}{2-3i}+\frac{5-i}{2+3i}\right) \times \frac{a}{b}=1\)[/tex], then [tex]\(a=2\)[/tex] and [tex]\(b=1\)[/tex].
