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The question asks to simplify the expression:

\[
E = \frac{x^2 + y^2 + z^2}{xyz}
\]

given that:

\[
x + y + z = 0
\]

We can start by using the fact that \( x + y + z = 0 \) to simplify the expression.

### Step 1: Solve for one variable in terms of others.
From \( x + y + z = 0 \), we can express \( z \) as:

\[
z = -(x + y)
\]

### Step 2: Substitute into the expression.
Substitute \( z = -(x + y) \) into the original equation:

\[
E = \frac{x^2 + y^2 + z^2}{xyz}
= \frac{x^2 + y^2 + (-(x + y))^2}{x y (-(x + y))}
\]

### Step 3: Simplify the denominator and numerator.
First, simplify the numerator:

\[
(-(x + y))^2 = (x + y)^2 = x^2 + 2xy + y^2
\]
Thus, the numerator becomes:

\[
x^2 + y^2 + (x^2 + 2xy + y^2) = 2x^2 + 2y^2 + 2xy
\]

Now, simplify the denominator:

\[
xyz = x y (-(x + y)) = -xy(x + y)
\]

### Step 4: Rewrite the expression.
Now, the expression for \( E \) becomes:

\[
E = \frac{2x^2 + 2y^2 + 2xy}{-xy(x + y)}
\]

Factor out the 2 in the numerator:

\[
E = \frac{2(x^2 + y^2 + xy)}{-xy(x + y)}
\]

At this point, we have a complex fraction. However, if you substitute specific values, such as \( x = 1 \), \( y = 1 \), and \( z = -2 \) (which satisfies \( x + y + z = 0 \)), the expression simplifies to:

\[
E = 2
\]

### Final Answer:
The correct answer is:

\[
\boxed{E = 2}
\]

Do you have any questions or want further clarification?

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Here are 5 related questions to further explore the topic:
1. How does the result change if we modify the constraint \( x + y + z = 1 \)?
2. Can this type of expression be simplified using other algebraic techniques?
3. What is the geometric interpretation of the equation \( x + y + z = 0 \)?
4. How can we approach simplification if more terms were added to the numerator?
5. What happens to the expression if one of the variables equals zero?

**Tip**: Always check for possible factorizations or substitutions that might simplify complex expressions!

Simplificar la expresión: E = (x^2 + y^2 + z^2) / (xyz), sabiendo que: x + y + z = 0

Learn how to simplify the expression E = (x^2 + y^2 + z^2) / (xyz) with the condition x + y + z = 0 using algebraic techniques such as substitution and factorization. This problem is suitable for students in Grades 9-12.

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