To find the exact solutions of the quadratic equation \(x^2 - 5x - 1 = 0\), we use the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
In this equation, \(a = 1\), \(b = -5\), and \(c = -1\).
Using these values, we can follow these steps:
1. Identify the coefficients:
[tex]\[
a = 1, \quad b = -5, \quad c = -1
\][/tex]
2. Calculate the discriminant:
[tex]\[
\Delta = b^2 - 4ac = (-5)^2 - 4 \cdot 1 \cdot (-1) = 25 + 4 = 29
\][/tex]
3. Substitute the coefficients and the discriminant into the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-(-5) \pm \sqrt{29}}{2 \cdot 1} = \frac{5 \pm \sqrt{29}}{2}
\][/tex]
Therefore, the exact solutions of the equation \(x^2 - 5x - 1 = 0\) are:
[tex]\[
x = \frac{5 \pm \sqrt{29}}{2}
\][/tex]
This corresponds to the solution:
[tex]\[
x = \frac{5 \pm \sqrt{29}}{2}
\][/tex]
