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Q.
If two roots of the equation $x^3-p x^2+q x-r=0$ are equal in magnitude but opposite in sign, then:
Complex Numbers and Quadratic Equations
Solution:
Let the roots be $\alpha, \beta,-\beta$
then $\alpha+\beta-\beta=p$
$\Rightarrow \alpha=p$.....(1)
and $ \alpha \beta-\alpha \beta-\beta^2=q$
$\Rightarrow \beta^2=-q$..........(2)
also $-\alpha \beta^2=r$
$\Rightarrow p q=r $ [using (1)].