Question

If one root of a quadratic equation with rational coefficient is $\left(\begin{array}{c}\frac{3\sqrt{5}}{\sqrt{10}+\sqrt{20}+\sqrt{40}-\sqrt{5}-\sqrt{80}}\end{array}\right)$ then quadratic equation is

• A. $x^2+2x-1=0$
• B. $x^2-2x-1=0$
• C. $x^2-2\sqrt{2}+1=0$
• D. $x^2+2x-3=0$

Answer 1

The correct option is B $x^2-2x-1=0$

$\alpha =\frac{3\sqrt{5}}{\sqrt{10}+\sqrt{20}+\sqrt{40}-\sqrt{5}-\sqrt{80}}$
$=\frac{3\sqrt{5}}{\sqrt{10}+2\sqrt{5}+2\sqrt{10}-\sqrt{5}-4\sqrt{5}}$
$\frac{3\sqrt{5}}{3\sqrt{10}-3\sqrt{5}}=\frac{\sqrt{5}}{\sqrt{10}-\sqrt{5}}\times \frac{\sqrt{10}+\sqrt{5}}{\sqrt{10}+\sqrt{5}}$
$\frac{\sqrt{50}+5}{5}=\frac{5\sqrt{2}+5}{5}$
$=\sqrt{2}+1$
$\beta =1-\sqrt{2}$
Sum (s) $=\alpha +\beta =2$
Product(p) $F=\alpha \beta =\left(\begin{array}{c}1+\sqrt{2}\end{array}\right)\left(\begin{array}{c}1-\sqrt{2}\end{array}\right)$
$=1-2=-1$
Equation $=x^2-sx+p=0$
$=x^2-2x-1$