Question:

Find the values of $k$ so that the quadratic equation $x^{2}-4 k x+k=0$ has equal roots.

Solution:

It is given that the quadratic equation $x^{2}-4 k x+k=0$ has equal roots.

$\therefore D=0$

$\Rightarrow(-4 k)^{2}-4 \times 1 \times k=0$

$\Rightarrow 16 k^{2}-4 k=0$

$\Rightarrow 4 k(4 k-1)=0$

$\Rightarrow k=0$ or $4 k-1=0$

$\Rightarrow k=0$ or $k=\frac{1}{4}$

Hence, 0 and $\frac{1}{4}$ are the required values of $k$.