Questions: sqrt(x) + sqrt(x-5) = 1

Solution Steps

To solve the equation \(\sqrt{x} + \sqrt{x-5} = 1\), we can follow these steps:

1. Isolate one of the square root terms.
2. Square both sides of the equation to eliminate the square root.
3. Simplify and solve the resulting equation.
4. Check the solutions to ensure they satisfy the original equation.

Starting with the equation: \[ \sqrt{x} + \sqrt{x - 5} = 1 \] we can isolate one of the square root terms: \[ \sqrt{x - 5} = 1 - \sqrt{x} \]

Next, we square both sides to eliminate the square root: \[ x - 5 = (1 - \sqrt{x})^2 \] Expanding the right side gives: \[ x - 5 = 1 - 2\sqrt{x} + x \]

Rearranging the equation, we have: \[ x - 5 = 1 - 2\sqrt{x} + x \] Subtracting \(x\) from both sides results in: \[ -5 = 1 - 2\sqrt{x} \] Adding 5 to both sides yields: \[ 0 = 6 - 2\sqrt{x} \] Rearranging gives: \[ 2\sqrt{x} = 6 \] Dividing by 2: \[ \sqrt{x} = 3 \] Squaring both sides results in: \[ x = 9 \]

We need to verify if \(x = 9\) satisfies the original equation: \[ \sqrt{9} + \sqrt{9 - 5} = 3 + 2 = 5 \neq 1 \] Since this does not hold, we conclude that there are no valid solutions.

Final Answer