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}
]
