\(f\left(a\right)=\sqrt{25a^2-30a+9}\)

               \(g\left(a\right)=a\)

We have the following:

        \(f\left(a\right)=g\left(a\right)+7\)

\(\Leftrightarrow\sqrt{25a^2-30a+9}=a+7\)

\(\Leftrightarrow\left\{{}\begin{matrix}25a^2-30a+9=\left(a+7\right)^2\\a+7\ge0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}a\ge-7\\24a^2-44a-40=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}a\ge-7\\\left[{}\begin{matrix}a=\dfrac{5}{2}\\a=-\dfrac{2}{3}\end{matrix}\right.\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}a=\dfrac{5}{2}\\a=-\dfrac{2}{3}\end{matrix}\right.\)

Finally, there are two values of a.