X^2=5-12i

Solution for X^2=5-12i equation:

X^2=5-12X

We move all terms to the left:

X^2-(5-12X)=0

We add all the numbers together, and all the variables

X^2-(-12X+5)=0

We get rid of parentheses

X^2+12X-5=0

a = 1; b = 12; c = -5;
Δ = b2-4ac
Δ = 122-4·1·(-5)
Δ = 164
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:

$\sqrt{\Delta}=\sqrt{164}=\sqrt{4*41}=\sqrt{4}*\sqrt{41}=2\sqrt{41}$
$X_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(12)-2\sqrt{41}}{2*1}=\frac{-12-2\sqrt{41}}{2}
$
$X_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(12)+2\sqrt{41}}{2*1}=\frac{-12+2\sqrt{41}}{2}
$