Find the values of $k$ for which the following equations have real roots.
$x^{2} - 4 k x + k = 0$

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Solution

$x^{2} - 4 k x + k = 0$
This is in the form of $a x^{2} + b x + c = 0$
Here $a = 1, b = - 4 k, c = k$
For the equation to have real roots ,
discriminant $\geq 0 ⟹ (- 4 k)^{2} - 4 (1) (k) \geq 0$
$⟹ k (4 k - 1) \geq 0 ⟹ k \in (- \infty, 0] \cup [1 / 4, \infty)$

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x^{2} + 4 k x + (k^{2} - k + 2) = 0.

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