To solve the quadratic equation \( x^{2} - 4x - 5 = 0 \), we can use the quadratic formula, which is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In this equation, \( a = 1 \), \( b = -4 \), and \( c = -5 \).
Now, let's calculate the discriminant \( b^2 - 4ac \):
\[ b^2 = (-4)^2 = 16 \]
\[ 4ac = 4 \cdot 1 \cdot (-5) = -20 \]
\[ b^2 - 4ac = 16 - (-20) = 16 + 20 = 36 \]
Now we plug this back into the quadratic formula:
\[ x = \frac{-(-4) \pm \sqrt{36}}{2 \cdot 1} \]
\[ x = \frac{4 \pm 6}{2} \]
Now, we can find the two solutions:
-
\( x = \frac{4 + 6}{2} = \frac{10}{2} = 5 \)
-
\( x = \frac{4 - 6}{2} = \frac{-2}{2} = -1 \)
Thus, the solutions to the equation \( x^{2} - 4x - 5 = 0 \) are:
\[ x = 5 \quad \text{and} \quad x = -1 \]