Question

Find the number of solution(s) of the equation $\sqrt{6-4x-x^2}=x+4$.

Answer 1

We have $\sqrt{6-4x-x^2}=x+4$
This equation is equivalent to the system
$\{\begin{array}{c}x+4\ge 0\\ 6-4x-x^2={(x+4)}^2\end{array}\Rightarrow \{\begin{array}{c}x\ge -4\\ x^2+6x+5=0\end{array}$
Solving the equation $x^2+6x+5=0$,
we find that from $x_1=-1$ and $x_2=-5$ only $x_1=-1$ satisfies the condition $x\ge -4$ .
Consequently, the number $-1$ is the only solution of the given equation .