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.
For the equation \(x^2 + x^\frac{1}{2} - 37 = 0\), find two consecutive integers between which a root of the equation lies.
Using the method given in the first part of the question, calculate an approximation to the root, giving three significant figures in your answer.
