Question

If one root of the $x^2-x-k=0$ be the 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

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

Let the root be $a$ and $a^2$

$\therefore a+a^2=\frac{-1}{1}=1$

and $a\times a^2=\frac{-k}{1}=-k$

$\Rightarrow (a+a^2{)}^3=(1{)}^3$

$\Rightarrow a^3+a^6+3a.a^2(a+a^2)=1$

$\Rightarrow (-k)+(-k{)}^2+3(-k\left)\right(-1)=1$

$\Rightarrow k^2-4k-1=0$

$\Rightarrow k=\frac{4\pm \sqrt{16+4}}{2}=2\pm \sqrt{5}$