2x+sqrt(x+14)=8 solve for x.

Rearrange as 

√(x + 14) = 8 - 2x

 

Square both sides

x + 14 = (8 - 2x)²

x + 14 = 64 - 32x + 4x²

 

Collect like terms

4x² - 33x + 50 = 0

$${\mathtt{4}}{\mathtt{\,\times\,}}{{\mathtt{x}}}^{{\mathtt{2}}}{\mathtt{\,-\,}}{\mathtt{33}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,\small\textbf+\,}}{\mathtt{50}} = {\mathtt{0}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\frac{{\mathtt{25}}}{{\mathtt{4}}}}\\
{\mathtt{x}} = {\mathtt{2}}\\
\end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{6.25}}\\
{\mathtt{x}} = {\mathtt{2}}\\
\end{array} \right\}$$

 

Check in the original equation:

$${\mathtt{LHS}} = {\sqrt{{\mathtt{6.25}}{\mathtt{\,\small\textbf+\,}}{\mathtt{14}}}} \Rightarrow {\mathtt{LHS}} = {\mathtt{4.5}}$$

$${\mathtt{RHS}} = {\mathtt{8}}{\mathtt{\,-\,}}{\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{6.25}} \Rightarrow {\mathtt{RHS}} = -{\mathtt{4.5}}$$

So x ≠ 6.25

 

Check the other solution

$${\mathtt{LHS}} = {\sqrt{{\mathtt{2}}{\mathtt{\,\small\textbf+\,}}{\mathtt{14}}}} \Rightarrow {\mathtt{LHS}} = {\mathtt{4}}$$

$${\mathtt{RHS}} = {\mathtt{8}}{\mathtt{\,-\,}}{\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{2}} \Rightarrow {\mathtt{RHS}} = {\mathtt{4}}$$

So x = 2 is the solution to the original equation.