Given:
[tex]\frac{5+2i}{3+4i}[/tex]Required:
We need to divide 5+2i by 3+4i.
Explanation:
The conjugate of 3+4i is 3-4i.
Multiply and divide the given expression by 3-4i
[tex]=\frac{(5+2\imaginaryI)(3-4i)}{(3+4\imaginaryI)(3-4i)}[/tex][tex]Use\text{ }(a+b)(a-b)=a^2-b^2.[/tex][tex]=\frac{5(3-4i)+2\imaginaryI(3-4i)}{(3^2-(4i)^2)}[/tex][tex]=\frac{5(3)-5(4i)+2\imaginaryI(3)-2i(4i)}{9-(-16)}[/tex][tex]=\frac{15-20i+6i-8(-1)}{9+16}[/tex][tex]=\frac{15-14i+8}{25}[/tex][tex]=\frac{23-14i}{25}[/tex]Final answer:
[tex]\frac{23-14i}{25}[/tex]
