To identify the solutions of the equation [tex]\(\sqrt{x-8} + 8 = x\)[/tex], let's examine each given value step by step.
Step 1: Check [tex]\( x = 9 \)[/tex]
Substitute [tex]\( x = 9 \)[/tex] into the equation:
[tex]\[
\sqrt{9-8} + 8 = 9
\][/tex]
Simplify inside the square root:
[tex]\[
\sqrt{1} + 8 = 9
\][/tex]
Since [tex]\(\sqrt{1} = 1\)[/tex], we have:
[tex]\[
1 + 8 = 9
\][/tex]
This is a true statement. Therefore, [tex]\( x = 9 \)[/tex] is a solution.
Step 2: Check [tex]\( x = 8 \)[/tex]
Substitute [tex]\( x = 8 \)[/tex] into the equation:
[tex]\[
\sqrt{8-8} + 8 = 8
\][/tex]
Simplify inside the square root:
[tex]\[
\sqrt{0} + 8 = 8
\][/tex]
Since [tex]\(\sqrt{0} = 0\)[/tex], we have:
[tex]\[
0 + 8 = 8
\][/tex]
This is also a true statement. Therefore, [tex]\( x = 8 \)[/tex] is also a solution.
Conclusion:
Since both [tex]\( x = 9 \)[/tex] and [tex]\( x = 8 \)[/tex] satisfy the equation [tex]\(\sqrt{x-8} + 8 = x\)[/tex], the correct answer is:
[tex]\[
x = 8 \text{ and } x = 9
\][/tex]
