m^2-6m-11=0

To solve the quadratic equation \( m^2 - 6m - 11 = 0 \), we can use the quadratic formula, which is given by:

\[ m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

In this equation, \( a = 1 \), \( b = -6 \), and \( c = -11 \).

Now, let's calculate the discriminant (\( b^2 - 4ac \)):

\[ b^2 - 4ac = (-6)^2 - 4 \times 1 \times (-11) = 36 + 44 = 80 \]

Next, we can substitute these values into the quadratic formula:

\[ m = \frac{-(-6) \pm \sqrt{80}}{2 \times 1} = \frac{6 \pm \sqrt{80}}{2} \]

We can simplify \( \sqrt{80} \):

\[ \sqrt{80} = \sqrt{16 \times 5} = 4\sqrt{5} \]

Now substituting back into our equation:

\[ m = \frac{6 \pm 4\sqrt{5}}{2} \]

Dividing both the terms in the numerator by 2 gives us:

\[ m = 3 \pm 2\sqrt{5} \]

Thus, the solutions for \( m \) are:

\[ m = 3 + 2\sqrt{5} \quad \text{and} \quad m = 3 - 2\sqrt{5} \]