The problem is to solve the equation $k + \sqrt{k^2 + k - 2} = 2$ for the variable $k$.

1. Problem Description

The problem is to solve the equation $k + \sqrt{k^2 + k - 2} = 2$ for the variable $k$.

2. Solution Steps

First, isolate the square root term:

$\sqrt{k^2 + k - 2} = 2 - k$

Square both sides of the equation to eliminate the square root:

$(\sqrt{k^2 + k - 2})^2 = (2 - k)^2$

$k^2 + k - 2 = 4 - 4k + k^2$

Now, simplify the equation:

$k^2 + k - 2 = k^2 - 4k + 4$

$k - 2 = -4k + 4$

$5k = 6$

$k = \frac{6}{5}$

We need to check if this solution is valid by substituting $k = \frac{6}{5}$ back into the original equation:

$\frac{6}{5} + \sqrt{(\frac{6}{5})^2 + \frac{6}{5} - 2} = 2$

$\frac{6}{5} + \sqrt{\frac{36}{25} + \frac{30}{25} - \frac{50}{25}} = 2$

$\frac{6}{5} + \sqrt{\frac{16}{25}} = 2$

$\frac{6}{5} + \frac{4}{5} = 2$

$\frac{10}{5} = 2$

$2 = 2$

The solution is valid.

3. Final Answer

$k = \frac{6}{5}$