To determine which equation is equivalent to [tex]\(\sqrt{x^2 + 81} = x + 10\)[/tex], follow these steps:
1. Start 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]
3. Simplify both sides of the resulting equation:
- The left-hand side simplifies as follows:
[tex]\[
(\sqrt{x^2 + 81})^2 = x^2 + 81
\][/tex]
- The right-hand side simplifies as follows:
[tex]\[
(x + 10)^2 = x^2 + 20x + 100
\][/tex]
4. Combine these simplified expressions to form the new equation:
[tex]\[
x^2 + 81 = x^2 + 20x + 100
\][/tex]
Therefore, the equation equivalent to [tex]\(\sqrt{x^2 + 81} = x + 10\)[/tex] is:
[tex]\[
x^2 + 81 = x^2 + 20x + 100
\][/tex]
So, the correct answer is:
[tex]\[
x^2 + 81 = x^2 + 20x + 100
\][/tex]
