Question

The roots of the equation ${(x-\frac{1}{x})}^2=4+\frac{3}{2}(x-\frac{1}{x})$ are

• A. $\pm 1,2,\frac{1}{2}$
• B. $\pm 1,2,-\frac{1}{2}$
• C. $\pm 1,-2,-\frac{1}{2}$
• D. None of these

Answer 1

The correct option is D None of these

Let
$x-\frac{1}{x}=t$
Hence,
$t^2=4+\frac{3t}{2}$
$t^2-\frac{3t}{2}-4=0$
$(t-\frac{3}{4}{)}^2-\frac{9}{16}-\frac{64}{16}=0$
$(t-\frac{3}{4}{)}^2=\frac{73}{16}$
Hence
$t=\frac{3\pm \sqrt{73}}{4}$
$x-\frac{1}{x}=\frac{3\pm \sqrt{73}}{4}$
$x^2-\left(\frac{3+\sqrt{73}}{4}\right)x-1=0$ and $x^2-\left(\frac{3-\sqrt{73}}{4}\right)x-1=0$
Hence, answer is none of these .