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**PythagorasNeophyte** Question: show that the roots of the equation $\displaystyle (px-1)^2 +3px -5 = 0$ are real and distinct for all real values of $\displaystyle p$ and $\displaystyle p \neq 0$

My workings:

$\displaystyle (px-1)^2 +3px -5 = 0$

$\displaystyle p^2x^2 -2px +1 +3px -5 = 0$

$\displaystyle p^2x^2 +px -4= 0$

Here $\displaystyle a = p^2, b = p, c = -4$. Now

$\displaystyle b^2 -4ac = p^2 -4p^2(-4)$

$\displaystyle = p^2 + 16p^2$

$\displaystyle = 17p^2$

This doesn't prove anything. Can you please help me? I don't have anyone to guide me.