Finding roots of complex quadratic equation

Question

I'm trying to solve for the following equation:

$$|(1+50*i*x)^2|$$ I keep getting the form $$-2500x^2 + 100ix + 1 $$ when the problem needs to have the following form:

$$2500x^2 + 1$$

What steps are necessary for me to take to reach the latter response?

Answer 1

recall that $|a+bi|^2=|(a+bi)^2|$, because $$|a+bi|^2=(\sqrt{a^2+b^2})^2=a^2+b^2$$ and $$|(a+bi)^2|=|a^2-b^2+2abi|=\sqrt{(a^2-b^2)^2+4a^2b^2}=a^2+b^2$$ So $$|(1+50xi)^2|=|1+50xi|^2=(\sqrt{1+(50x)^2})^2=2500x^2+1$$

Answer 2

For a complex number $z=a+bi$, where $a,b$ are real numbers, the value $|z|$ is defined as $\sqrt{a^2+b^2}$. The value $|z^2|$ is equal to $|z|^2$, does this answer your question?

Answer 3

$|1+50ix|^2=(1+50ix)(1-50ix)=2500x^2+1$,

you just needed to take the conjugate, but this is assuming that $x$ is real.