Question

Solve:$\sqrt{x-2}+2=x$.

Short answer

The solution is$x=2,3$.

Step-by-step solution

Step 1. Given Information  

We are given the quadratic equation $\sqrt{x-2}+2=x$and we need to solve it.

Step 2. Finding the solution  

Isolating the radical on one side of the equation:

$$\begin{gathered} \sqrt{x-2}+2=x \\ \Rightarrow \sqrt{x-2}+2-2=x-2 \\ \Rightarrow \sqrt{x-2}=x-2 \end{gathered}$$

Squaring both sides of the equation, we get:

$$\begin{gathered} {\left(\sqrt{x-2}\right)}^2={\left(x-2\right)}^2 \\ \Rightarrow x-2=x^2-4x+4 \\ \Rightarrow x^2-4x+4-x+2=0 \\ \Rightarrow x^2-5x+6=0 \\ \Rightarrow x^2-2x-3x+6=0 \\ \Rightarrow x\left(x-2\right)-3\left(x-2\right)=0 \\ \Rightarrow \left(x-3\right)\left(x-2\right)=0 \end{gathered}$$

Using the zero product property, we get:

$$\begin{gathered} \left(x-2\right)=0 \\ \Rightarrow x=2 \\ \left(x-3\right)=0 \\ \Rightarrow x=3 \end{gathered}$$

Step 3. Checking the answer  

$$\begin{gathered} x=2, \\ \sqrt{x-2}+2=x \\ \Rightarrow \sqrt{2-2}+2=2 \\ \Rightarrow 0+2=2 \\ \Rightarrow 2=2 \\ x=3, \\ \sqrt{3-2}+2=3 \\ \Rightarrow \sqrt{1}+2=3 \\ \Rightarrow 1+2=3 \\ \Rightarrow 3=3 \end{gathered}$$