The question involves solving for the value of $k$ that minimizes the expression $x_1^2 + x_2^2$, where $x_1$ and $x_2$ are the roots of the quadratic equation:

$x^2 + kx + k = 0$

Let’s go through the steps to solve it.

Step 1: Use Vieta’s Formulas

For a quadratic equation of the form $x^2 + kx + k = 0$, Vieta's formulas give us:

• The sum of the roots: $x_1 + x_2 = -k$
• The product of the roots: $x_1 \cdot x_2 = k$

Step 2: Minimize the Sum of Squares

We are asked to minimize $x_1^2 + x_2^2$. We can use the identity:

$x_1^2 + x_2^2 = (x_1 + x_2)^2 - 2x_1x_2$

Substituting from Vieta’s formulas:

$x_1^2 + x_2^2 = (-k)^2 - 2k = k^2 - 2k$

Step 3: Minimize the Expression

To minimize $k^2 - 2k$, we can take the derivative with respect to $k$ and set it equal to zero:

$\frac{d}{dk}(k^2 - 2k) = 2k - 2$

Setting this equal to zero:

$2k - 2 = 0 \quad \Rightarrow \quad k = 1$

Thus, the value of $k$ that minimizes $x_1^2 + x_2^2$ is $k = 1$.

Conclusion

The correct answer is $k = 1$, which corresponds to the last option in the list.

Would you like further details, or do you have any questions?

Here are 5 related questions:

1. How do you solve quadratic equations using Vieta's formulas?
2. Can you explain other methods to find the minimum of a quadratic expression?
3. How does completing the square relate to minimizing expressions like this?
4. What are the general properties of the roots of a quadratic equation?
5. How do quadratic equations change when coefficients vary?

Tip: For any quadratic equation, using Vieta's relations simplifies handling sums and products of roots without explicitly solving the equation.