Find the square root of
`2x-1+2sqrt(x^(2)-x-6)`.

Answer

To find the square root of the expression \( 2x - 1 + 2\sqrt{x^2 - x - 6} \), we will follow these steps:

### Step 1: Simplify the expression under the square root
We start with the expression:
\[ 2x - 1 + 2\sqrt{x^2 - x - 6} \]
First, we need to simplify \( \sqrt{x^2 - x - 6} \).

### Step 2: Factor the quadratic expression
The quadratic \( x^2 - x - 6 \) can be factored. We look for two numbers that multiply to \(-6\) and add to \(-1\). These numbers are \(-3\) and \(2\):
\[ x^2 - x - 6 = (x - 3)(x + 2) \]
Thus,
\[ \sqrt{x^2 - x - 6} = \sqrt{(x - 3)(x + 2)} \]

### Step 3: Substitute back into the expression
Now we substitute this back into our original expression:
\[ 2x - 1 + 2\sqrt{(x - 3)(x + 2)} \]

### Step 4: Rewrite the expression
We can rewrite the expression as:
\[ \sqrt{2x - 1 + 2\sqrt{(x - 3)(x + 2)}} \]

### Step 5: Recognize the form of a perfect square
We can express \( 2x - 1 + 2\sqrt{(x - 3)(x + 2)} \) in the form of a perfect square:
\[ \sqrt{( \sqrt{x - 3} + \sqrt{x + 2} )^2} \]
This gives us:
\[ \sqrt{( \sqrt{x - 3} + \sqrt{x + 2} )^2} = \sqrt{x - 3} + \sqrt{x + 2} \]

### Final Answer
Thus, the square root of the expression \( 2x - 1 + 2\sqrt{x^2 - x - 6} \) is:
\[ \sqrt{x - 3} + \sqrt{x + 2} \]