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  4. In the quadratic equation x2+(p+i q) x+3 i=0, p q are real. If the sum of the squares of the roots is 8 then :

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Q. In the quadratic equation $x^2+(p+i q) x+3 i=0, p \& q$ are real. If the sum of the squares of the roots is 8 then :

Complex Numbers and Quadratic Equations

Solution:

$x^2+(p+i q) x+3 i=0$
$\alpha+\beta=-(p+i q), \alpha \beta=3 i$
$\alpha^2+\beta^2=(\alpha+\beta)^2-2 \alpha \beta=[-(p+i q)]^2-6 i$
$=\left(p^2-q^2\right)+i(2 p q-6)=8$
$\Rightarrow p^2-q^2=8$ and $p q=3$
$\Rightarrow p=3, q=1$ or $p=-3, q=-1$