\alpha is a root of the equation px^2+2qx+r=0 or not

Question

α

is a root of the equation

p x^{2} + 2 q x + r = 0

or not where the following determinant is given 0

Answer 1

Step 1
Your determinant value seems to be wrong. The correct determinant is
$- α^{2} p^{2} r + α^{2} p q^{2} - 2 α p q r + 2 α q^{3} - p r^{2} + q^{2} r$
$= - (p r - q^{2}) (α^{2} p + 2 q α + r)$
so unless $p r = q^{2}, α$ is indeed a root of $p x^{2} + 2 q x + r$

\alpha is a root of the equation px^2+2qx+r=0 or not

Answers (1)

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