To solve the equation [tex]\(\sqrt{x-4} + 5 = 2\)[/tex], we need to isolate the square root term and analyze if there is any possible value of [tex]\(x\)[/tex] that satisfies the equation. Let's go through the steps in detail:
1. Isolate the square root term:
[tex]\[
\sqrt{x-4} + 5 = 2
\][/tex]
Subtract 5 from both sides of the equation:
[tex]\[
\sqrt{x-4} = 2 - 5
\][/tex]
Simplify the right-hand side:
[tex]\[
\sqrt{x-4} = -3
\][/tex]
2. Analyze the result:
Here, we have [tex]\(\sqrt{x-4} = -3\)[/tex]. Recall that the square root of a real number is always non-negative. Therefore, there is no real number [tex]\(x\)[/tex] such that [tex]\(\sqrt{x-4}\)[/tex] would equal [tex]\(-3\)[/tex].
3. Conclusion:
Since it is impossible for the square root to be negative, there are no real values of [tex]\(x\)[/tex] that satisfy the equation [tex]\(\sqrt{x-4} + 5 = 2\)[/tex].
Thus, the correct answer is:
no solution
