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  4. The quadratic equation with real co-efficients one of whose complex roots has the real part 12 and modulus 13 is:

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Q. The quadratic equation with real co-efficients one of whose complex roots has the real part 12 and modulus 13 is:

Complex Numbers and Quadratic Equations

Solution:

Let the roots be $12+\alpha i$; other roots is $12-\alpha i$ now $z \overline{ z }=| z |^2$
$\therefore 144+ a ^2=169 \Rightarrow \alpha= \pm 5 $
$\text { Hence roots are } 12 \pm 5 i $