Solve the Radical Equation: Find X in x + √x = -√x

To solve the equation $x + \sqrt{x} = -\sqrt{x}$, we follow these steps:

• Step 1: Simplify the equation by combining like terms. We have: $x + \sqrt{x} = -\sqrt{x} \Rightarrow x + 2\sqrt{x} = 0$
• Step 2: Isolate $x$ by rearranging terms: $x = -2\sqrt{x}$
• Step 3: Square both sides to eliminate the square root: $x^2 = 4x$
• Step 4: Rearrange the equation into standard quadratic form: $x^2 - 4x = 0$
• Step 5: Factor the quadratic equation: $x(x - 4) = 0$
• Step 6: Solve for $x$: $x = 0 \quad \text{or} \quad x = 4$

Step 7: Verify each solution in the original equation. We find:

• For $x = 0$: $0 + \sqrt{0} = -\sqrt{0}$ $0 = 0 \quad \text{True}$
• For $x = 4$: $4 + \sqrt{4} = -\sqrt{4}$ $4 + 2 \neq -2 \quad \text{False}$

Thus, the only valid solution is $x = 0$.

Therefore, the solution to the equation is $x = 0$, which corresponds to choice 2.