Find all real solutions of the equation. sqrt{4x+2}=sqrt{x+2}

To solve the equation \(\sqrt{4x+2}=\sqrt{x+2}\), we begin by squaring both sides to eliminate the square roots: \[ 4x + 2 = x + 2. \] Next, we rearrange the equation by subtracting \(x\) and 2 from both sides: \[ 4x + 2 - x - 2 = 0 \implies 3x = 0. \] Dividing both sides by 3 gives: \[ x = 0. \] Now, we need to verify that this solution satisfies the original equation. Plugging \(x = 0\) back into the equation: \[ \sqrt{4(0)+2} = \sqrt{0+2} \implies \sqrt{2} = \sqrt{2}. \] Since both sides are equal, \(x = 0\) is indeed a solution. Finally, we need to check for any extraneous solutions. Since we squared the equation, it's possible we introduced new solutions that don't actually satisfy the original equation. However, in this case, \(x = 0\) is the only solution we found, and it works. Thus, the real solution to the equation is: \[ \boxed{0}. \]