Write \(x^2 + 6kx +144\) in the form \((x+p)^2+q\) and thus obtain expressions for \(p\) and \(q\) in terms of \(k\).
Hence find the range of values for \(k^2\) such that \(x^2+6kx+144\) is positive for all values of \(x\), and deduce the corresponding range of values for \(k\).
When drawing the graph of \(y = (x+p)^2+q\), what do \(p\) and \(q\) represent?
How does knowing \(p\) and \(q\) help us to determine whether the quadratic function is always positive?
Here is the graph of \(y = x^2+6kx+144\).