Need help with intro question about complex polynomials

Hello everyone I am having some difficulties understanding some concepts. I am trying to solve for the roots of a complex polynomial.

$$f(x)=(3i+1)x^{2}+(-6i-2)x+12$$

I tried to use the quadratic formula that is,

$$x= \frac{-(-2-6i) \pm \sqrt{(-6i-2)^{2}-4(3i+1)(12)}}{2(3+i)}$$

$$x= \frac{2+6i \pm \sqrt{-176-24i}}{6+2i}$$

Now is the point where I am confused, $$\sqrt{-176-24i}$$ which I rewrote as $$\sqrt{((-1)(8)(22+3i)}$$

and using what I thought were valid rules, wrote that as $$\sqrt{8}i\sqrt{22+3i}=2\sqrt{2}i\sqrt{22+3i}$$

so with that form I thought I could write,

$$x= \frac{2(1+3i \pm \sqrt{2}i\sqrt{22+3i})}{2(3+i)}$$

$$x= \frac{(1+3i \pm \sqrt{2}i\sqrt{22+3i})}{(3+i)}$$

But now I am stuck and am not sure if the form is correct, and I am not sure how I can proceed to get a final answer. Any help please? I am still stuck on this. Is there a way I can do it without changing into polar form? Also I appreciate all the hints for finding the square roots and such, but I am still not even sure my work is correct. Could someone help with that? It seems to disagree with wolfram.

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In general, let $z=x+iy$ where $x$ and $y$ are real-valued numbers. We can express $z$ in Polar Coordinates as $z=\sqrt{x^2+y^2}e^{i\arctan(x,y)}$ where the Arctangent Function $\arctan(x,y)$ is given by

$$
\arctan(x,y)=\begin{cases}
\arctan(y/x)&,x>0\\\\
\pi+\arctan(y/x)&,x<0,y>0\\\\
-\pi+\arctan(y/x)&,x<0,y<0\\\\
\pi/2&,x=0,y>0\\\\
-\pi/2&,x=0,y<0
\end{cases}$$

Then, the square root of $z$, $z^{1/2}$ is one of the two values

$$z^{1/2}=\pm (x^2+y^2)^{1/4}e^{i\arctan(x,y)/2} \tag 1$$

We can convert $z^{1/2}$ back to rectangular coordinates using Euler’s Identity on $(1)$. Proceeding accordingly gives

$$\begin{align}
z^{1/2}&=\pm (x^2+y^2)^{1/4}\left(\cos\left(\frac12 \arctan(x,y)\right)+i\sin\left(\frac12\arctan(x,y)\right)\right)\\\\
&=\pm\sqrt{\frac{\sqrt{x^2+y^2}+x}{2}}\pm i\,\text{sgn}(y)\sqrt{\frac{\sqrt{x^2+y^2}-x}{2}}\tag 2
\end{align}$$

where in arriving at $(2)$ we used the Half-Angle Formulae for the sine and cosine funcitons.


Applying $(2)$ to the specific problem at hand for which $z=22+3i$, we have

$$\begin{align}
z^{1/2}&=\pm\left(\sqrt{\frac{\sqrt{(22)^2+(3)^2}+22}{2}}+i\text{sgn}(3)\sqrt{\frac{\sqrt{(22)^2+(3)^2}-22}{2}}\right)\\\\
&\approx 4.701255327631890
+i0.319063717127566
\end{align}$$

We’re looking for $a+ib$ such that
$$(a+ib)^2=a^2-b^2+2abi=-176-24i.$$
That is $a^2-b^2=-176$ and $2ab=-24$.
So $b=-12/a$ and $$a^4+176a^2-144=0,$$ which gives $$a^2=-88\pm\sqrt{7888}$$ hence $a=\pm 2\sqrt{-22+\sqrt{493}}.$
Now you can easily find $b$.

Another nice formula for the complex square root of some $z\in \Bbb C$ is
$$
\sqrt{z}=\pm\sqrt{|z|}·\frac{z+|z|}{\bigl|z+|z|\bigr|}
$$
encoding the facts that the root is on the angle bisector of $z$ and the real axis and that the absolute value is the root of the absolute value of $z$.