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  4. Let α and β are the roots of the equation x2-(p-2) x-p-1=0(p ∈ R). If α2+β2 is least, then p equals

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Q. Let $\alpha$ and $\beta$ are the roots of the equation $x^2-(p-2) x-p-1=0(p \in R)$. If $\alpha^2+\beta^2$ is least, then $p$ equals

Complex Numbers and Quadratic Equations

Solution:

$E=\alpha^2+\beta^2=(\alpha+\beta)^2-2 \alpha \beta=(p-2)^2+2(p+1)=p^2-2 p+6=(p-1)^2+5$
$E _{\min .}=5$ when $p =1$.