Question

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

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

Answer 1

Answer: (C)

Let the roots of the equation $x^2-x-k=0$ are $\alpha$ and ${\alpha }^2$ . Then, $\alpha +{\alpha }^2=1$ ..(i) and $\alpha .{\alpha }^2={\alpha }^3=-k$ $\Rightarrow$ $\alpha ={(-k)}^{1/3}$ On putting this value of a in Eq. (i), we get ${(-k)}^{1/3}+{(-k)}^{2/3}=1$ ?(ii) On cubing both sides, we get $(-k)+{(-k)}^2+3k[{(-k)}^{1/3}+{(-k)}^{2/3}]=1$ $\Rightarrow$ $-k+k^2-3k(k^{2/3}-k^{1/3})=1$ $\Rightarrow$ $k^2-k-3k(1)=1$ [Using Eq.(ii)] $\Rightarrow$ $k^2-4k-1=0$ $\Rightarrow$ $lk=\frac{4\pm \sqrt{20}}{2}=2\pm \sqrt{5}$