91Ó°ÊÓ

When solving quadratic equations, the first step is often to isolate the variable term. Here the equation is given as \( r^{2} - 64 = 0 \). To isolate the term containing the variable \( r^{2} \), we need to get rid of the constant '64'. This is done by adding 64 to both sides of the equation:

\( r^{2} - 64 + 64 = 0 + 64 \)

This simplifies the equation to:

\( r^{2} = 64 \)

Now, the variable term \( r^{2} \) is isolated, making it easier to solve for 'r'.

Isolating the variable term simplifies the equation and lets you focus on solving for the variable. This step is crucial in ensuring your solution is correct.

Once the variable term is isolated, the next step is to take the square root of both sides of the equation. Here, from the isolated term, we have:

\( r^{2} = 64 \)

To solve for 'r', take the square root on both sides:

\( r = \pm \sqrt{64} \)

This results in two possible solutions:

\( r = 8 \) and \( r = -8 \)

Remember that taking the square root gives both positive and negative root values because squaring either a positive or a negative number will yield the same positive number. Understanding this concept is crucial because it ensures that both potential solutions are considered in the final answer.

After solving for the variable, it's important to check if the solutions need rounding. In this case, our solutions were \( r = 8 \) and \( r = -8 \), both of which are whole numbers.

However, if the solutions were irrational (non-terminating and non-repeating decimals), rounding them to a specific decimal place, such as to the nearest thousandth, would be necessary. For example, if instead, the solutions were not whole numbers, you would round:\
\[ 2.6457513110645906 \approx 2.646 \]
As a rule of thumb, to round to the nearest thousandth, you look at the digit in the thousandth place and the digit immediately to its right. If that digit is 5 or greater, increase the thousandth place digit by one; if it's less than 5, leave the thousandth place digit as it is.

This check is a vital final step to ensure your results are presentable and accurate, adhering to the problem’s requirements.