Answer
The x-intercept of the graph of $f$ is $(\displaystyle \frac{4}{25},0)$
Work Step by Step
$f(x)=5x+13\sqrt{x}-6$
The x-intercept is the x-coordinate of a point where a line,
curve, or surface intersects the x-axis.
Substitute $f(x)$ for $0$ in the given function to find x-intercepts.
$ 5x+13\sqrt{x}-6=0\qquad$...substitute $\sqrt{x}$ for $u$ so that $u^{2}=x$.
$ 5u^{2}+13u-6=0\qquad$... solve with the Quadratic formula.
$u=\displaystyle \frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$
$u=\displaystyle \frac{-13\pm\sqrt{169+120}}{10}$
$u=\displaystyle \frac{-13\pm\sqrt{289}}{10}$
$u=\displaystyle \frac{-13\pm 17}{10}$
$u=\displaystyle \frac{-13+17}{10}=\frac{4}{10}=\frac{2}{5}$ or $u=\displaystyle \frac{-13-17}{10}=\frac{-30}{10}=-3$
Bring back $\sqrt{x}=u$.
$\displaystyle \sqrt{x}=\frac{2}{5}$ or $\sqrt{x}=-3$
$\sqrt{x}$ is a positive number which is why we discard $-3$.
$\displaystyle \sqrt{x}=\frac{2}{5}$
$x=\displaystyle \frac{4}{25}$
The x-intercept of the graph of $f$ is $(\displaystyle \frac{4}{25},0)$
