Equation with roots adjusted by reciprocal squares

The roots of the quadratic equation \(x^2-5x+2=0\) are \(\alpha\) and \(\beta\).

1. Write down the values of \(\alpha+\beta\) and of \(\alpha\beta\).
2. Without solving the equation, find the value of each of the following.
   1. \(\dfrac{1}{\alpha^2}+\dfrac{1}{\beta^2}\)
   2. \(\left(\alpha-\dfrac{3}{\beta^2}\right)\left(\beta-\dfrac{3}{\alpha^2}\right)\)
3. Determine the quadratic equation with integer coefficients which has roots \(\alpha-\dfrac{3}{\beta^2}\) and \(\beta-\dfrac{3}{\alpha^2}\).