X^2-(3+2i)*X+5+i=0

Solution for X^2-(3+2i)*X+5+i=0 equation:

X^2-(3+2X)*X+5+X=0

We add all the numbers together, and all the variables

X^2-(2X+3)*X+X+5=0

We add all the numbers together, and all the variables

X^2+X-(2X+3)*X+5=0

We multiply parentheses

X^2-2X^2+X-3X+5=0

We add all the numbers together, and all the variables

-1X^2-2X+5=0

a = -1; b = -2; c = +5;
Δ = b2-4ac
Δ = -22-4·(-1)·5
Δ = 24
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$X_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$X_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{24}=\sqrt{4*6}=\sqrt{4}*\sqrt{6}=2\sqrt{6}$
$X_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-2)-2\sqrt{6}}{2*-1}=\frac{2-2\sqrt{6}}{-2}
$
$X_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-2)+2\sqrt{6}}{2*-1}=\frac{2+2\sqrt{6}}{-2}
$