To solve the equation \( r^2 - 75 = 0 \), follow these steps:

### Solution

1. Start with the equation:
   \[
   r^2 - 75 = 0
   \]

2. Add \( 75 \) to both sides to isolate \( r^2 \):
   \[
   r^2 = 75
   \]

3. Take the square root of both sides to solve for \( r \):
   \[
   r = \pm \sqrt{75}
   \]

4. Simplify \( \sqrt{75} \):
   \[
   \sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3}
   \]

5. So the solutions for \( r \) are:
   \[
   r = \pm 5\sqrt{3}
   \]

### Final Answer
\[
r = 5\sqrt{3} \quad \text{or} \quad r = -5\sqrt{3}
\]

Would you like additional details on any of these steps?

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Here are five related questions for further exploration:
1. How would you solve an equation like \( r^2 - k = 0 \) for any constant \( k \)?
2. What are the solutions to \( r^2 + 49 = 0 \)?
3. How do you simplify square roots like \( \sqrt{75} \) in general?
4. What is the difference between rational and irrational numbers in solutions?
5. Can the square root of any number be simplified further, and how do you know?

**Tip:** Remember, when solving quadratic equations, if the squared term equals a positive constant, take both the positive and negative roots.

Learn how to solve the equation r^2 - 75 = 0 with a step-by-step solution, including isolating the variable, using square roots, and simplifying terms. Suitable for students in Grades 9-10, covering key algebraic concepts and the square root property.

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