First, let's identify the coefficients of the quadratic equation [tex]\( x^2 - 5x - 1 = 0 \)[/tex]. Here, [tex]\( a = 1 \)[/tex], [tex]\( b = -5 \)[/tex], and [tex]\( c = -1 \)[/tex].
The general form for solving a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Let's go through the steps to find the discriminant:
[tex]\[
\text{discriminant} = b^2 - 4ac
\][/tex]
Substitute [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
\text{discriminant} = (-5)^2 - 4(1)(-1) = 25 + 4 = 29
\][/tex]
Now, substitute [tex]\( b = -5 \)[/tex], the discriminant, and [tex]\( a = 1 \)[/tex] into the quadratic formula:
[tex]\[
x = \frac{-(-5) \pm \sqrt{29}}{2(1)} = \frac{5 \pm \sqrt{29}}{2}
\][/tex]
Thus, the solutions to the quadratic equation [tex]\( x^2 - 5x - 1 = 0 \)[/tex] are:
[tex]\[
x = \frac{5 + \sqrt{29}}{2} \quad \text{and} \quad x = \frac{5 - \sqrt{29}}{2}
\][/tex]
Comparing this with the options provided, we find that the correct answer is:
D. [tex]\( x = \frac{5 \pm \sqrt{29}}{2} \)[/tex]
