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- Prove that the roots of the equation x^2 - (a + b) x + ab - p^2 = 0 are real for all real values of a, b and p.
- Show that (k^2)x^2 + 2 = kx - x^2 has non-real roots for all real values for k.
- If $x = \dfrac{3-\sqrt{a}}{\sqrt{2}}$ and $y = \dfrac{4 +\sqrt{a}}{\sqrt{2}}$ , determine the value of $(x+y)^{2}$
- If $f(x)=0$ has roots $x=\dfrac{-5\pm \sqrt{3 -12k^2}}{4}$, for which values of $k$ will the roots be equal?
- Show that the roots of the following expression are real and rational for all values of k.

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