Answer
$\left( \frac{4}{9},0 \right)$
Work Step by Step
$f\left( x \right)=3x+10\sqrt{x}-8$.
The x-intercept occurs when $f\left( x \right)=0$,
$3x+10\sqrt{x}-8=0$ ……(1)
Let $u=\sqrt{x}$ and ${{u}^{2}}=x$,
Substitute the values of $u$ and ${{u}^{2}}$ in equation (1),
$3{{u}^{2}}+10u-8=0$
Now, factor the equation.
$\left( 3u-2 \right)\left( u+4 \right)=0$
If $\left( 3u-2 \right)=0$:
$\begin{align}
& 3u-2=0 \\
& 3u=2 \\
& u=\frac{2}{3}
\end{align}$
If $\left( u+4 \right)=0$:
$\begin{align}
& u+4=0 \\
& u=-4 \\
\end{align}$
Now, replace $u$ with $\sqrt{x}$
$\begin{align}
& 3x+10\sqrt{x}-8=0 \\
& 3\left( \frac{4}{9} \right)+10\left( \frac{2}{3} \right)-8=0 \\
& \frac{4}{3}+\frac{20}{3}-8=0 \\
& \frac{24}{3}-8=0
\end{align}$
$\begin{align}
& \frac{24}{3}-8=0 \\
& 8-8=0
\end{align}$
So, $x=\frac{4}{9}$
Thus, the x-intercept is $\left( \frac{4}{9},0 \right)$.
