To solve the equation \( x^2 - 5x - 1 = 0 \), we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
For the equation \( x^2 - 5x - 1 \):
- \( a = 1 \)
- \( b = -5 \)
- \( c = -1 \)
Now, plug in these values into the formula:
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Calculate \( b^2 - 4ac \): \[ b^2 = (-5)^2 = 25 \] \[ 4ac = 4 \cdot 1 \cdot (-1) = -4 \] \[ b^2 - 4ac = 25 - (-4) = 25 + 4 = 29 \]
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Substitute back into the quadratic formula: \[ x = \frac{-(-5) \pm \sqrt{29}}{2 \cdot 1} \] \[ x = \frac{5 \pm \sqrt{29}}{2} \]
Thus, the solutions to the equation are:
\[ x = \frac{5 \pm \sqrt{29}}{2} \]
This corresponds to option D:
D. \( x = \frac{5 \pm \sqrt{29}}{2} \)
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