It is known that the equation \(f(x) = 0\) has only one real root in the interval \(a < x < b\) and that \(f(a) < 0\), \(f(b) > 0\). The two points \(A\) and \(B\) lie on the graph of \(f(x)\) and have coordinates \((a,f(a))\) and \((b,f(b))\) respectively. By finding where the line \(AB\) cuts the axis of \(x\), show that \[\begin{equation*} \frac{af(b) - bf(a)}{f(b) - f(a)} \end{equation*}\]

is an approximation to the root.

How could we find where the line \(AB\) cuts the \(x\)-axis?

You may find the resource In-betweens helpful here.