Answer
$x=\frac{1-\sqrt{241}}{6}\approx -2.42 $, $x=\frac{1+\sqrt{241}}{6}\approx 2.75 $
Work Step by Step
Given \begin{equation}
\sqrt{4 x^2+9 x+5}=5+x.
\end{equation} First square both sides of the radical equation to eliminate the radical sign. Rearrange and solve for $x$.
\begin{equation}
\begin{aligned}
\sqrt{4 x^2+9 x+5}&=5+x\\
\left( \sqrt{4 x^2+9 x+5}\right)^2&= \left( 5+x\right)^2\\
4 x^2+9 x+5& = 25+10x+x^2\\
3x^2-x-20& = 0.
\end{aligned}
\end{equation} Use the quadratic formula to find the value(s) of $x$.
\begin{equation}
\begin{aligned}
a &= 3\ , b= -1\ ,\ c = -20\\
x & = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\\
x&=\frac{-(-1) \pm \sqrt{(-1)^2-4 \cdot 3 \cdot(-20)}}{2 \cdot 3} \\
&=\frac{1\pm\sqrt{241}}{6}\\
\implies x& = \frac{1-\sqrt{241}}{6}\\
& = -2.42069\\
x& = \frac{1+\sqrt{241}}{6}\\
& = 2.75402
\end{aligned}
\end{equation} Check. \begin{equation}
\begin{aligned}
&\sqrt{4\cdot (-2.42069)^2+9\cdot(-2.42069)+5}\\
& \stackrel{?}{=} 5-2.42069 \\
2.58& =2.58\quad \textbf{True}\\
&\sqrt{4\cdot (2.75402)^2+9\cdot(2.75402)+5}\\
& \stackrel{?}{=} 5+2.75402 \\
7.75& =7.75\quad \textbf{True}
\end{aligned}
\end{equation} The solution is $$x=\frac{1-\sqrt{241}}{6}\approx -2.42, x=\frac{1+\sqrt{241}}{6}\approx 2.75 $$
