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  4. Two roots of the cubic equation x 3-5 x 2- ax +45=0 are real and additive inverse of each other then a is equal to

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Q. Two roots of the cubic equation $x ^3-5 x ^2- ax +45=0$ are real and additive inverse of each other then a is equal to

Complex Numbers and Quadratic Equations

Solution:

Let the roots are $\alpha,-\alpha, \beta$
$\text { S.O.R. }=\alpha-\alpha+\beta=5 \Rightarrow \beta=5 $
$\text { P.O.R. }=\alpha(-\alpha) \beta=-45 \Rightarrow \alpha^2=9 $
$\Rightarrow \alpha=3 $
$\text { Now, }- a =3(-3)+3(5)+3(-5) $
$- a =-9 $
$a =9 $