To solve the equation \( x^2 - 4x - 6 = 0 \), we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1 \), \( b = -4 \), and \( c = -6 \).
First, we calculate the discriminant \( b^2 - 4ac \):
\[ b^2 - 4ac = (-4)^2 - 4 \cdot 1 \cdot (-6) = 16 + 24 = 40 \]
Now we can plug the values into the quadratic formula:
\[ x = \frac{-(-4) \pm \sqrt{40}}{2 \cdot 1} = \frac{4 \pm \sqrt{40}}{2} \]
The square root of 40 can be simplified:
\[ \sqrt{40} = \sqrt{4 \cdot 10} = 2\sqrt{10} \]
So, our equation becomes:
\[ x = \frac{4 \pm 2\sqrt{10}}{2} \]
Now, we simplify further:
\[ x = 2 \pm \sqrt{10} \]
Thus, the solutions to the equation \( x^2 - 4x - 6 = 0 \) are:
\[ x = 2 + \sqrt{10} \quad \text{and} \quad x = 2 - \sqrt{10} \]