I got an question to find the product of the roots of $x^2 + 18x + 30 = 2\sqrt{x^2 + 18x + 45}$. This is what I did:
$y = x^2 + 18x$. So $y + 30 = 2\sqrt{y + 45}$.
Squaring it on both sides:
$(y + 30)^2 = (2\sqrt{y + 45})^2\Rightarrow y^2 + 30^2 + 60y = 4(y + 45)\Rightarrow y^2 + 900 + 60y - 4y - 180 = 0\Rightarrow y^2 + 720 + 56y = 0$.
Then using quadratic formula I got the values of $y$ as $-20$ and $-36$. Now plugging In the values as $x^2 + 18x + 36/+20 = 0$, I don't get a whole number as the value for $x$. This is the place I'm stuck.
Any help is appreciated
The product of the (real or complex) roots of $x^2+18x+36$ is $36$, the product of those of $x^2+18x+20$ is $20$, hence the product of the roots of $(x^2+18x+36)(x^2+18x+20)$ is $720$.
There would remain to eliminate the roots which do not belong to the domain of validity of the initial equation, the roots of which must satisfy: $$x^2+18x+30\ge0.$$