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 \(\frac{3k^2}{4}\).