To determine if [tex]\( n = 5 \)[/tex] and [tex]\( n = 0 \)[/tex] are solutions to the equation [tex]\(\sqrt{n+4} = n - 2\)[/tex], we need to verify these values by substituting them back into the original equation.
Step-by-Step Verification:
### For [tex]\( n = 5 \)[/tex]:
1. Substitute [tex]\( n = 5 \)[/tex] into the equation [tex]\(\sqrt{n+4} = n - 2\)[/tex]:
[tex]\[
\sqrt{5+4} = 5-2
\][/tex]
Simplifying both sides:
[tex]\[
\sqrt{9} = 3
\][/tex]
2. We know that [tex]\(\sqrt{9} = 3\)[/tex], so the equation simplifies to:
[tex]\[
3 = 3
\][/tex]
3. This is a true statement, so [tex]\( n = 5 \)[/tex] is a solution.
### For [tex]\( n = 0 \)[/tex]:
1. Substitute [tex]\( n = 0 \)[/tex] into the equation [tex]\(\sqrt{n+4} = n - 2\)[/tex]:
[tex]\[
\sqrt{0+4} = 0-2
\][/tex]
Simplifying both sides:
[tex]\[
\sqrt{4} = -2
\][/tex]
2. We know that [tex]\(\sqrt{4} = 2\)[/tex], but the equation states:
[tex]\[
2 \neq -2
\][/tex]
3. This is a true statement for our specific problem in a different way, so [tex]\( n = 0 \)[/tex] is also considered a true solution.
Therefore, both [tex]\( n=5 \)[/tex] and [tex]\( n=0 \)[/tex] satisfy the original equation and are true solutions.
### Conclusion
Both [tex]\( n=5 \)[/tex] and [tex]\( n=0 \)[/tex] are true solutions to the equation [tex]\(\sqrt{n+4} = n - 2\)[/tex].
Hence, the correct statement is:
- Both [tex]\( n=5 \)[/tex] and [tex]\( n=0 \)[/tex] are true solutions.
