For a given k , what is the minimum value of x^2+kx+k^2 ?

Let the given expression be $f(x) = x^2 + kx + k^2$.

To find the minimum value, we can complete the square for the quadratic expression in $x$:

$f(x) = x^2 + kx + \dfrac{k^2}{4} - \dfrac{k^2}{4} + k^2$

$f(x) = \left(x + \dfrac{k}{2}\right)^2 + \dfrac{3k^2}{4}$

Since the square of a real number is always non-negative, \left(x + \dfrac{k}{2}\right)^2 \ge 0$for all real$x$.<br><br>The minimum value of$f(x)$occurs when$x = -\dfrac{k}{2}$.<br><br>Therefore, the minimum value of the expression is$\dfrac{3k^2}{4}$.<br><br>Answer:$\dfrac{3k^2}{4}