Question

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

Answer 1

We have, $x^{2}+18x+30=2\sqrt{x^{2} + 18 x + 45}$
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 $20.$