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  4. A quadratic equation whose one root is 2 cos 2 (π/10) is

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Q. A quadratic equation whose one root is $2 \cos ^2 \frac{\pi}{10}$ is

Complex Numbers and Quadratic Equations

Solution:

Let one root be $\alpha$
$\therefore \alpha=2 \cos ^2 18^{\circ}=1+\cos 36^{\circ}=1+\frac{(\sqrt{5}+1)}{4}=\frac{5+\sqrt{5}}{4} . $
$\therefore \text { other root }=\beta=\frac{5-\sqrt{5}}{4}$
$\therefore \alpha+\beta=\frac{5}{2} \text { and } \alpha \beta=\frac{20}{16}=\frac{5}{4}$
$\therefore$ Quadratic equation is $4 x^2-10 x+5=0$.