Answer
$x=\dfrac{3-i}{2}
\text{ OR }
x=\dfrac{3+i}{2}$
Work Step by Step
The given equation, $
2x(x-3)=-5
,$ is equivalent to
\begin{align*}
2x^2-6x&=-5
&\text{ (use Distributive Property)}
\\
2x^2-6x+5&=0
.\end{align*}
Using $ax^2+bx+c=0,$ the equation above has $a=
2
,$ $b=
-6
,$ and $c=
5
.$ Using the Quadratic Formula which is given by $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a},$ then
\begin{align*}\require{cancel}x&=
\dfrac{-(-6)\pm\sqrt{(-6)^2-4(2)(5)}}{2(2)}
\\\\&=
\dfrac{6\pm\sqrt{36-40}}{4}
\\\\&=
\dfrac{6\pm\sqrt{-4}}{4}
\\\\&=
\dfrac{6\pm\sqrt{-1}\cdot\sqrt{4}}{4}
\\\\&=
\dfrac{6\pm\sqrt{-1}\cdot2}{4}
\\\\&=
\dfrac{6\pm i\cdot2}{4}
&\text{ (use $i=\sqrt{-1}$)}
\\\\&=
\dfrac{6\pm 2i}{4}
\\\\&=
\dfrac{\cancel6^3\pm \cancel2^1i}{\cancel4^2}
&\text{ (divide by $2$)}
\\\\&=
\dfrac{3\pm i}{2}
.\end{align*}
The solutions are $
x=\dfrac{3-i}{2}
\text{ and }
x=\dfrac{3+i}{2}
.$
