Question

Find the product of real roots of equation, $x^2+18x+30=2\sqrt{x^2+18x+45}.$

Answer 1

Answer: 20

We have, $x^2+18x+30=2\sqrt{x^2+18x+45}$
Let $x^2+18x+30=t$
Therefore,
$t=2\sqrt{t+15}$
$\Rightarrow t^2-4t-60=0$
$\Rightarrow t^2-10t+6t-60=0$
$\Rightarrow \left(t-10\right)\left(t+6\right)=0$
$\Rightarrow t=10,-6$
Put $x^2+18x+30=10$
$\Rightarrow x^2+18x+20=0$
So, product $P=20$
and $t=x^2+18x+30=-6<0$ (Not possible)
Hence, product of roots is $20.$