To determine which equation is equivalent to [tex]\(\sqrt{x^2 + 81} = x + 10\)[/tex], let's carefully undergo a detailed, step-by-step transformation of the given equation.
1. Starting with the given equation:
[tex]\[\sqrt{x^2 + 81} = x + 10\][/tex]
2. Square both sides to eliminate the square root:
[tex]\[
(\sqrt{x^2 + 81})^2 = (x + 10)^2
\][/tex]
This simplifies to:
[tex]\[
x^2 + 81 = (x + 10)^2
\][/tex]
3. Expand the right side:
[tex]\[
x^2 + 81 = x^2 + 20x + 100
\][/tex]
Thus, the equation that is equivalent to [tex]\(\sqrt{x^2 + 81} = x + 10\)[/tex] is:
[tex]\[ x^2 + 81 = x^2 + 20x + 100 \][/tex]
So, the correct choice is:
[tex]\[ \boxed{x^2 + 81 = x^2 + 20 x + 100} \][/tex]
