Question

Let one root of the quadratic equation $x^2-x-k=0$ be square of the other. Then find the value of $k$.

Answer 1

Let the roots of the quadratic equation $x^2-x-k=0$ be $\alpha ,{\alpha }^2$.

Then from the relation between the roots and co-efficients we get,

$\alpha +{\alpha }^2=1$......(1) and ${\alpha }^3=-k$......(2).

From (1) we get,

${\alpha }^2+\alpha -1=0$

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

Let, $\alpha =\frac{-1+\sqrt{5}}{2}$

${\alpha }^3=\frac{-1-15+3\sqrt{5}+5\sqrt{5}}{8}$

or, ${\alpha }^3=-2+\sqrt{5}$.

Similarly when $\alpha =\frac{-1-\sqrt{5}}{2}$ then ${\alpha }^3=-2-\sqrt{5}$.

So, we have ${\alpha }^3=-2\pm \sqrt{5}$.

From (2) we have $k=2\pm \sqrt{5}$.