Question

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

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

Answer 1

The correct option is A $2\pm \sqrt{5}$

Let $\alpha$ and ${\alpha }^2$ be the roots of $x^2-x-k=0$. Then,
$\alpha +{\alpha }^2=1\dots \dots \left(1\right)$
and ${\alpha }^3=-k\Rightarrow \alpha =-k^{1/3}$
Substituting in $\left(1\right)$ we have
$(-k{)}^{1/3}+(-k{)}^{2/3}=1$
or $(k^{2/3}-k^{1/3})=1$
Taking cube on both sides, we get
$k^2-k-3k(k^{2/3}-k^{1/3})=1$
$k^2-k-3k\left(1\right)=1$
$k^2-4k-1=0$
$\Rightarrow k=2\pm \sqrt{5}$
Hence, option 'A' is correct.