Answer
$\{1-2i,1+2i\}$
Work Step by Step
The given equation has $a=1,b=-2$ and $c=5$.
The discriminant is
$=b^2-4ac$
$=(-2)^2-4(1)(5)$
$=4-20$
$=-16$
Use Quadratic formula.
$x=\dfrac{-b\pm \sqrt {b^2-4ac}}{2a}$
Substitute the values of $a, b, c, $ and the discriminant.
$x=\dfrac{-(-2)\pm \sqrt {(-16)}}{2(1)}$
Simplify.
$x=\dfrac{2\pm \sqrt{16(-1)}}{2}$
$x=\dfrac{2\pm 4i}{2}$
Factor out $2$.
$x=\dfrac{2(1\pm 2i)}{2}$
Cancel the common factor $2$.
$x=1\pm 2i$
The equation has two solutions: $1-2i$ and $1+2i$.
Check:
For $x=1-2i$.
$=(1-2i)^2-2(1-2i)+5$
$=1-4i+4i^2-2+4i+5$
$=4+4i^2$
$=4-4$
$=0$
For $x=1+2i$.
$=(1+2i)^2-2(1+2i)+5$
$=1+4i+4i^2-2-4i+5$
$=4+4i^2$
$=4-4$
$=0$
Hence, the solution set is $\{1-2i,1+2i\}$.
