To solve the equation [tex]\(\sqrt{x - 4} + 5 = 2\)[/tex], we will follow these steps:
1. Isolate the square root:
[tex]\[\sqrt{x - 4} + 5 = 2\][/tex]
Subtract 5 from both sides:
[tex]\[\sqrt{x - 4} = 2 - 5\][/tex]
Simplify the right-hand side:
[tex]\[\sqrt{x - 4} = -3\][/tex]
2. Analyze the square root:
The left-hand side of the equation is a square root, [tex]\(\sqrt{x - 4}\)[/tex]. By definition, the square root of a number is always non-negative. This implies:
[tex]\[\sqrt{x - 4} \geq 0\][/tex]
However, we have [tex]\(\sqrt{x - 4} = -3\)[/tex], which is a contradiction since [tex]\(\sqrt{x - 4}\)[/tex] cannot be negative. Thus, this equation has no valid real solution.
3. Evaluate the given choices:
The given choices for [tex]\(x\)[/tex] are:
- [tex]\(x = -17\)[/tex]
- [tex]\(x = 13\)[/tex]
- [tex]\(x = 53\)[/tex]
None of these values will satisfy the equation [tex]\(\sqrt{x - 4} + 5 = 2\)[/tex] because the equation itself leads to an impossible scenario (square root equating to a negative number).
Therefore, the final conclusion is that there is [tex]\(\boxed{\text{no solution}}\)[/tex].
