7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7de3f581-eff1-4671-87a9-55ca1bb97890-20_591_962_312_548}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
Figure 1 shows a sketch of the curve with equation \(y = \left| x ^ { 2 } - 8 \right|\) and a sketch of the straight line with equation \(y = m x + c\), where \(m\) and \(c\) are positive constants.
The equation
$$\left| x ^ { 2 } - 8 \right| = m x + c$$
has exactly 3 roots, as shown in Figure 1.
- Show that
$$m ^ { 2 } - 4 c + 32 = 0$$
Given that \(c = 3 m\)
- determine the value of \(m\) and the value of \(c\)
- Hence solve
$$\left| x ^ { 2 } - 8 \right| \geqslant m x + c$$