Question

if one root is equal To the square of the other root of the equation x²+x-k=0 what is the value of k.

Answer 1

Heya

Let the roots be :
$y \: \: || \: \: {y}^{2}$

Sum of roots :
$y + {y}^{2} = - 1$

$= > {y}^{2} + y + 1 = 0$

$= > y = \omega \: || \: { \omega}^{2}$
$\omega \: is \: cube \: root \: of \: unity$

$= > { \omega}^{3} = 1$

Now, coming back to the original equation :
${x}^{2} + x - k = 0$
Since, y and y^2 are roots, the product of zeroes = -k

$= > y( {y}^{2} ) = - k$

$= > {y}^{3} = - k$
But y = omega

$= > - k = 1$

And hence, the value of k = -1