Question

Solve the radical equation. Check all proposed solutions.
StartRoot x EndRootplusStartRoot x minus 5 EndRootequals1
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Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
The solution set is StartSet nothing EndSet
  
.
​(Simplify your answer. Type an integer or a simplified fraction. Use a comma to separate answers as​ needed.)
B.
The solution set is the empty set.

Answer 1

To solve the equation \(\sqrt{x} + \sqrt{x - 5} = 1\), we will first isolate one of the radicals and then square both sides to eliminate the square roots. Let's go through the steps.

1. Isolate one of the radicals:
\[
\sqrt{x - 5} = 1 - \sqrt{x}
\]

2. Square both sides:
\[
x - 5 = (1 - \sqrt{x})^2
\]

Expanding the right side:
\[
x - 5 = 1 - 2\sqrt{x} + x
\]

3. Rearranging gives:
\[
x - 5 = 1 + x - 2\sqrt{x}
\]
\[
-5 = 1 - 2\sqrt{x}
\]

Simplifying further:
\[
-6 = -2\sqrt{x}
\]

4. Divide by -2:
\[
3 = \sqrt{x}
\]

5. Square both sides to solve for \(x\):
\[
x = 9
\]

Now we have a potential solution \(x = 9\). We need to check if this value satisfies the original equation:

- Substitute \(x = 9\) back into the original equation:
\[
\sqrt{9} + \sqrt{9 - 5} = 3 + \sqrt{4} = 3 + 2 = 5 \neq 1
\]

Since the left-hand side equals 5 and not 1, this value does not satisfy the original equation.

Since we found a solution that doesn’t work in the original equation, the solution set is empty.

Now, select the correct choice from the options provided. The correct choice is:
B. The solution set is the empty set.