To identify the equation equivalent to [tex]\(\sqrt{x^2 + 81} = x + 10\)[/tex], let's go through each step methodically.
1. We start with the given equation:
[tex]\[
\sqrt{x^2 + 81} = x + 10
\][/tex]
2. To eliminate the square root, we square both sides of the equation:
[tex]\[
(\sqrt{x^2 + 81})^2 = (x + 10)^2
\][/tex]
3. Simplifying both sides, we have:
[tex]\[
x^2 + 81 = (x + 10)^2
\][/tex]
4. Now, we expand the right-hand side of the equation:
[tex]\[
x^2 + 81 = x^2 + 20x + 100
\][/tex]
5. When simplified, the equation becomes:
[tex]\[
x^2 + 81 = x^2 + 20x + 100
\][/tex]
Hence, the equation that is equivalent to [tex]\(\sqrt{x^2 + 81} = x + 10\)[/tex] is:
[tex]\[
x^2 + 81 = x^2 + 20x + 100
\][/tex]
Therefore, the correct answer is:
[tex]\[
x^2 + 81 = x^2 + 20 x + 100
\][/tex]
