https://file.aiquickdraw.com/mathbot/file/17267448324097b9p8agu.jpg

The equation given in the image is:

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

To find all the values of \(x\), \(y\), and \(z\), we can attempt various strategies including trial solutions or methods like completing the square. First, let's rearrange and attempt simplifications.

The equation is quadratic in terms of \(x\), \(y\), and \(z\), involving cross terms such as \(xy\) and \(yz\), and linear terms like \(6z\). I will attempt to factor or find a structured way to solve this step-by-step. Let me proceed with the algebra and calculations.

The solutions for the variables \(x\), \(y\), and \(z\) involve the following two expressions for \(x\), with \(y\) and \(z\) being arbitrary:

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

This means \(y\) and \(z\) can be any values, and \(x\) depends on both through the above relationship. To explore specific values, you'd need to assign values to \(y\) and \(z\), and then determine \(x\).

Would you like to explore specific values for \(y\) and \(z\), or do you have any other questions?

Here are some follow-up questions:
1. What values of \(y\) and \(z\) can simplify the expression for \(x\)?
2. Can the discriminant of the square root be non-negative for real solutions?
3. How does changing the values of \(z\) affect \(x\)?
4. Can you derive constraints on \(y\) and \(z\) for real solutions?
5. What happens when \(y = z = 0\)?

**Tip**: When dealing with quadratic expressions involving multiple variables, try to simplify cross terms by substituting known values to gain insights.

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

Discover how to solve the multivariable quadratic equation x^2 + 13y^2 + 2z^2 - 6xy - 4yz - 6z + 9 = 0. This solution involves advanced algebraic techniques, including factoring and methods for solving quadratic forms with multiple variables. Ideal for advanced high school or undergraduate students.

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