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)

$2\pm \sqrt{5}$

Let the roots of the equation $x^2-x-k=0$ be $a,a^2$.

Then from the relation between the roots and coefficents we get,

$a+a^2=1$ and $a^3=-k$.......(1).

or, $a^2+a-1=0$

or, $a=\frac{-1\pm \sqrt{5}}{2}$

Now $a^3=-2\mp \sqrt{5}$.

From (1) we get,

$k=2\pm \sqrt{5}$.