It is given that \[f(x)=(x-2)^2-\lambda(x+1)(x+2).\]

  1. Find the values of \(\lambda\) for which the equation \(f(x)=0\) has two equal roots.

Could we express \(f(x)\) in the form \(ax^2+bx+c\)?

What is the condition for an equation to have two equal roots?

  1. Show that, when \(\lambda=2\), \(f(x)\) has a maximum value of \(25\).

  2. Given that the curve \(y=f(x)\) has a turning point when \(x=\dfrac{1}{4}\), find the value of \(\lambda\) and sketch the curve for this value of \(\lambda\).

Here the curve \(y=f(x)\) is in red, while the line \(x = \frac{1}{4}\) is in blue.