To solve for \( r \) in the equation 

\[
n = \frac{1 + r^2}{2},
\]

follow these steps:

1. Multiply both sides by 2 to eliminate the fraction:

\[
2n = 1 + r^2.
\]

2. Subtract 1 from both sides:

\[
2n - 1 = r^2.
\]

3. Finally, take the square root of both sides to solve for \( r \):

\[
r = \sqrt{2n - 1} \quad \text{or} \quad r = -\sqrt{2n - 1}.
\]

Thus, the solutions for \( r \) are:

\[
r = \pm \sqrt{2n - 1}.
\]

If you have any further questions or need more details, feel free to ask! Here are some related questions:

1. What are the conditions for \( n \) for \( r \) to be a real number?
2. How does changing \( n \) affect the value of \( r \)?
3. Can you provide examples with specific values of \( n \)?
4. How would you graph the relationship between \( n \) and \( r \)?
5. What other mathematical concepts relate to this equation? 

**Tip:** When solving for variables in equations, always check the domain of the solution to ensure that it meets any necessary conditions.

Discover how to solve the equation n=\frac{1+r^2}{2} for r, including step-by-step instructions and related mathematical concepts. Learn about conditions for real solutions and the impact of variable changes. Suitable for students in Grades 7-9.

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