Solve x - Square Root of x = Square Root of x: Finding All Values

Let's solve the equation $x - \sqrt{x} = \sqrt{x}$.

Step 1: Start by simplifying the equation.
The equation is given by:

$x - \sqrt{x} = \sqrt{x}$

Step 2: Isolate the square root by adding $\sqrt{x}$ to both sides:

$x = 2\sqrt{x}$

Step 3: Square both sides to eliminate the square root.

$(x)^2 = (2\sqrt{x})^2$

This simplifies to:

$x^2 = 4x$

Step 4: Simplify and solve the quadratic equation:

Move all terms to one side:

$x^2 - 4x = 0$

Step 5: Factor the quadratic equation:

$x(x - 4) = 0$

Step 6: Solve for $x$ to find potential solutions:

$x = 0$ or $x = 4$.

Step 7: Check solutions by substituting back into the original equation:

• For $x = 0$:

$0 - \sqrt{0} = \sqrt{0}$ which simplifies to $0 = 0$. True.

• For $x = 4$:

$4 - \sqrt{4} = \sqrt{4}$ which simplifies to $4 - 2 = 2$. True.

Therefore, both solutions are valid.

The possible values of $x$ are 0 and 4.

Therefore, the solution to the problem is $0, 4$. Thus, the correct choice is option 2.