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

Step-by-step solution

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 + k^2 4 - k^2 4 + k^2 f(x) = (x + k 2 )^2 + 3k^2 4 Since the square of a real number is always non-negative, (x + k 2 )^2 0 for all real x . The minimum value of f(x) occurs when x = - k 2 . Therefore, the minimum value of the expression is 3k^2 4 . Answer: 3k^2 4