Question

What is the solution of x = 2+√(x-2)

Answer 1

We need to solve the equation:

`x=2+\sqrt[]{x-2}`

First, we can subtract 2 from both sides of the equation and then square both sides:

`\begin{gathered} x-2=2+\surd\mleft(x-2\mright)-2 \\ \\ x-2=\surd(x-2) \\ \\ (x-2)^2=\surd(x-2)^2 \\ \\ x^(2)-4x+4=x-2 \end{gathered}`

Now, we apply some operations on both sides to obtain:

`\begin{gathered} x^(2)-4x+4-x+2=x-2-x+2 \\ \\ x^(2)-5x+6=0 \end{gathered}`

And we can apply the quadratic formula to obtain the solutions. Notice, though, that x-2 cant be negative because this expression was originally inside the square root. So, we have the following condition for the solution:

`\begin{gathered} x-2\ge0 \\ \\ x\ge2 \end{gathered}`

Now, using the quadratic formula, we obtain:

`\begin{gathered} x=\frac{5\pm\sqrt[]{(-5)^(2)-4(1)(6)}}{2(1)} \\ \\ x=\frac{5\pm\sqrt[]{25-24}}{2} \\ \\ x=\frac{5\pm\sqrt[]{1}}{2} \\ \\ x=(5\pm1)/(2) \\ \\ x_1=(5-1)/(2)=2 \\ \\ x_2=(5+1)/(2)=3 \end{gathered}`

Therefore, since both solutions satisfy the condition imposed by the square root, the solutions are

x = 2 and x = 3