https://file.aiquickdraw.com/mathbot/file/1726744703091gk9b8yzo.jpg

The equation provided in the image is:

\[
x^2 + 13y^2 + 2z^2 - 6xy - 4yz - 6z + 9 = 0
\]

We need to find all the values of \(x\), \(y\), and \(z\) that satisfy this equation.

To solve this, we can attempt various algebraic methods such as substitution, factoring, or utilizing numerical methods if necessary. First, let's check for any possible simplifications or common factorings.

Let me proceed by solving this step by step.The solution to the given equation is expressed in terms of \(y\) and \(z\):

\[
x = 3y \pm \sqrt{-4y^2 + 4yz - 2z^2 + 6z - 9}
\]

This means that \(x\) depends on both \(y\) and \(z\). To find specific values for \(x\), \(y\), and \(z\), you would need additional conditions or constraints on the variables. However, these expressions give a general solution to the problem.

Would you like to explore this solution further by considering specific values for \(y\) and \(z\) or applying another method?

### Related Questions:
1. Can we simplify the discriminant further to get specific values of \(x\), \(y\), and \(z\)?
2. What happens when we substitute integer values for \(y\) and \(z\)?
3. Can this equation be solved using numerical methods instead?
4. How would this equation behave in terms of graphical representation?
5. Are there any restrictions on the values of \(y\) and \(z\) based on the square root?

**Tip:** Equations involving multiple variables can often be simplified or solved more easily when specific values or additional constraints are provided.

Find all the values of x, y, and z such that x^2 + 13y^2 + 2z^2 - 6xy - 4yz - 6z + 9 = 0.

Learn how to solve the multivariable quadratic equation x^2 + 13y^2 + 2z^2 - 6xy - 4yz - 6z + 9 = 0. This problem involves key algebraic concepts like solving for three variables, quadratic forms, and applying discriminants. Suitable for advanced students in Grades 10-12.

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