# $$\displaystyle\alpha$$ is a root of the equation

$\alpha$ is a root of the equation $p{x}^{2}+2qx+r=0$ or not where the following determinant is given 0
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Step 1
Your determinant value seems to be wrong. The correct determinant is
$-{\alpha }^{2}{p}^{2}r+{\alpha }^{2}p{q}^{2}-2 \alpha pqr+2 \alpha {q}^{3}-p{r}^{2}+{q}^{2}r$
$=-\left(pr-{q}^{2}\right)\left({\alpha }^{2}p+2 q\alpha +r\right)$
so unless  is indeed a root of $p{x}^{2}+2qx+r$