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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.

Jika x_1 dan x_2 akar-akar persamaan x^2 + kx + k = 0, maka x_1^2 + x_2^2 mencapai nilai minimum untuk k sama dengan.....

Learn how to solve for the value of k that minimizes x_1^2 + x_2^2, where x_1 and x_2 are the roots of the quadratic equation x^2 + kx + k = 0. This problem covers Vieta's formulas and quadratic equation minimization techniques. Suitable for students in Grades 9-11.

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