Answer
$x=3$
Work Step by Step
Using the properties of equality, the given equation, $
2x+\sqrt{x+1}=8
,$ is equivalent to
\begin{array}{l}\require{cancel}
\sqrt{x+1}=-2x+8
.\end{array}
Squaring both sides of the equation above, then the solution/s is/are
\begin{array}{l}\require{cancel}
x+1=(-2x)^2+2(-2x)(8)+(8)^2
\\\\
x+1=4x^2-32x+64
\\\\
-4x^2+(x+32x)+(1-64)=0
\\\\
-4x^2+33x-63=0
\\\\
4x^2-33x+63=0
\\\\
(4x-21)(x-3)=0
\\\\
x=\left\{ \dfrac{21}{4},3 \right\}
.\end{array}
Upon checking, only $
x=3
$ satisfies the original equation.