An algebra problem by Anandmay Patel

A function $f(x)$ is such that it outputs values by computing the inputs through some quadratic methods. Also, the output for a variable $x$ can be given in some form of a polynomial. Now, we know that $f(5)=0$ and $f(0)=5$ . The function gives the output as zero for only one (unique) input. The graph of the function touches the $x$ -axis at point $w$ and cuts the $y$ -axis at point $q$ . Find the $\textbf{length}$ of line $wq$ .

Note: In the answer options, $y = 0.5 \log_{10} 2 + \log_{10} 5$