Finding Minimum Value of Quadratic Expression

The problem asks for the minimum value of the quadratic expression $f(x) = x^2 + kx + k^2$ for a given constant $k$. Since the coefficient of the $x^2$ term (which is 1) is positive, the parabola opens upwards, meaning it has a minimum value.

Algebraic Method: Completing the Square

We can find the minimum value by completing the square for the expression $x^2 + kx + k^2$.

• Step 1: Rewrite the expression

  Start with the expression: $x^2 + kx + k^2$.

• Step 2: Complete the square

  To complete the square for $x^2 + kx$, we need to add and subtract $(\frac{k}{2})^2 = \frac{k^2}{4}$. $x^2 + kx + k^2 = \left( x^2 + kx + \frac{k^2}{4} \right) - \frac{k^2}{4} + k^2$ The terms inside the parenthesis form a perfect square:

  $ \left( x + \frac{k}{2} \right)^2 - \frac{k^2}{4} + k^2 $
• Step 3: Simplify the constant terms

  Combine the constant terms: $ \left( x + \frac{k}{2} \right)^2 + \frac{4k^2 - k^2}{4} $ $ \left( x + \frac{k}{2} \right)^2 + \frac{3k^2}{4} $

• Step 4: Determine the minimum value

  The term $\left( x + \frac{k}{2} \right)^2$ is always greater than or equal to 0, because it is a square. Its minimum value is 0, which occurs when $x = -\frac{k}{2}$. Therefore, the minimum value of the entire expression is obtained when $\left( x + \frac{k}{2} \right)^2 = 0$. Minimum Value = $0 + \frac{3k^2}{4} = \frac{3k^2}{4}$.

Conclusion

The minimum value of the expression $x^2 + kx + k^2$ is $\frac{3k^2}{4}$.