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The given equation is:

\[
y^2 - x^2 = 9, \quad x \text{ is a real number}.
\]

This equation represents a hyperbola. To solve for \( y \) in terms of \( x \), we can rewrite the equation as follows:

\[
y^2 = x^2 + 9.
\]

Now, take the square root of both sides:

\[
y = \pm\sqrt{x^2 + 9}.
\]

This solution gives the possible values of \( y \) for any real number \( x \). The hyperbola opens vertically because the \( y^2 \) term is positive.

Would you like further details on solving this or more information on hyperbolas?

Here are some related questions:
1. What is the shape of the graph of the given equation?
2. How does the equation change if \(x\) or \(y\) is restricted to positive values?
3. How can this equation be used to find points of intersection with other curves?
4. What happens if the 9 on the right side is replaced by another constant?
5. Can this equation represent a function of \(x\)? Why or why not?

**Tip:** Understanding the structure of conic sections like hyperbolas can help identify the behavior of complex equations involving squared terms.

Learn how to solve the hyperbola equation y^2 - x^2 = 9, where x is a real number. This problem covers key algebraic and geometric concepts, including the formula for hyperbolas and their graph. Suitable for students in Grades 9-12.

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