Find the values of k for which the following equations have real roots
$kx(x-2) + 6 = 0$

Given:

Given quadratic equation is $kx(x-2)+6= 0$.

To do:

We have to find the values of k for which the roots are real.

Solution:

$kx(x-2)+6=0$

$kx^2-2kx+6=0$

Comparing the given quadratic equation with the standard form of the quadratic equation $ax^2+bx+c=0$, we get,

$a=k, b=-2k$ and $c=6$.

The discriminant of the standard form of the quadratic equation $ax^2+bx+c=0$ is $D=b^2-4ac$.

$D=(-2k)^2-4(k)(6)$

$D=4k^2-24k$

The given quadratic equation has real roots if $D≥0$.

Therefore,

$4k^2-24k≥0$

$4k(k-6)≥0$

$4k≥0$ and $k-6≥0$

$k≥0$ and $k≥6$

This implies,

$k≥6$

The value of k is greater than or equal to 6.