Question

The quadratic equation whose one root is $\frac{3+\sqrt{5}}{2-\sqrt{5}}$ is

• A. $x^2+22x+4=0$
• B. $x^2+22x-4=0$
• C. $x^2+11x+8=0$
• D. $x^2+11x-8=0$

Answer 1

The correct option is B $x^2+22x-4=0$

One root $=\frac{3+\sqrt{5}}{2-\sqrt{5}}\frac{(2+\sqrt{5})}{(2+\sqrt{5})}$

$=\frac{6+3\sqrt{5}+2\sqrt{5}+5}{4-5}$

$=-(11+5\sqrt{5})=-11-5\sqrt{5}$

Other root $\Rightarrow -11+5\sqrt{5}$

Equation $\Rightarrow x^2-(-11-5\sqrt{5}-11+5\sqrt{5})x+(-11-5\sqrt{5})(-11+5\sqrt{5})=0$

$x^2+22x+(121-125)=0$

$x^2+22x-4=0$