Question

If one root of $x^2-x-k=0$ be square of the other, then $k=$

• A. $2\pm \sqrt{3}$
• B. $3\pm \sqrt{2}$
• C. $2\pm \sqrt{5}$
• D. $5\pm \sqrt{2}$

Answer 1

Answer: (C)

$2\pm \sqrt{5}$

Let $\alpha$ and $\beta$ be the roots of the given equation.
Hence,
$\beta ={\alpha }^2$
$\alpha +\beta =-(-1)=1$
$\alpha .\beta ={\alpha }^3=k$
Therefore,
$\alpha =k^{1/3}$
And
$\alpha +{\alpha }^2=1$
${\alpha }^2+\alpha -1=0$
$(\alpha +\frac{1}{2}{)}^2-\frac{1}{4}-1=0$
$(\alpha +\frac{1}{2}{)}^2=\frac{5}{4}$
$\alpha =\frac{1\pm \sqrt{5}}{2}$
Thus,
$k^{\frac{1}{3}}=\frac{1\pm \sqrt{5}}{2}$
Hence,
$k=2\pm \sqrt{5}$.