Answer
$x^2-8x+25=0$
Work Step by Step
The sum of the given roots, $
4-3i
\text{ and }
4+3i
,$ is
\begin{align*}
&
4-3i+4+3i
\\&=
(4+4)+(-3i+3i)
\\&=
8+0i
\\&=
8
.\end{align*}
Since the sum is given by the ratio $-\dfrac{b}{a},$ then
\begin{align*}
-\dfrac{b}{a}&=8
\\\\
-\dfrac{b}{1}&=8
&\text{ (given $a=1$)}
\\\\
-b&=8
\\
b&=-8
.\end{align*}
The product of the given roots is
\begin{align*}\require{cancel}
&
(4-3i)(4+3i)
\\&=
4(4)+4(3i)-3i(4)-3i(3i)
&\text{ (use FOIL)}
\\&=
16+12i-12i-9i^2
\\&=
16+12i-12i-9(-1)
&\text{ (use $i^2=-1$)}
\\&=
16+12i-12i+9
\\&=
(16+9)+(12i-12i)
\\&=
25+0i
\\&=
25
.\end{align*}
Since the product is given by the ratio $\dfrac{c}{a},$ then
\begin{align*}
\dfrac{c}{a}&=25
\\\\
\dfrac{c}{1}&=25
&\text{ (given $a=1$)}
\\\\
c&=25
.\end{align*}
Hence, the quadratic equation $ax^2+bx+c=0$ with the given roots is
\begin{align*}
x^2-8x+25=0
.\end{align*}
