Question

A quadratic equation with rational coefficients if one of its roots is ${\cot }^2{18}^{\circ }$ is

• A. $x^2+10x-5=0$
• B. $x^2+10x+5=0$
• C. $x^2-10x-5=0$
• D. $x^2-10x+5=0$

Answer 1

The correct option is D $x^2-10x+5=0$

${\cot }^2{18}^{\circ }=\frac{1+\cos {36}^{\circ }}{1-\cos {36}^{\circ }}=\frac{1+\frac{\sqrt{5}+1}{4}}{1-\frac{\sqrt{5}+1}{4}}=5+2\sqrt{5}$
Hence if $\alpha =5+2\sqrt{5}$ then $\beta =5-2\sqrt{5}$
$\alpha +\beta =10$; $\alpha \beta =25-20=5$
The quadratic equation is $x^2-10x+5=0$