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What are the solutions to the equation shown?



x squared minus 5 x minus 1 equals 0

Answer options with 4 options
A.
x equals StartFraction 5 plus or minus StartRoot 6 EndRoot over 2 EndFraction

B.
x equals StartFraction 5 plus or minus StartRoot 14 EndRoot over 2 EndFraction

C.
x equals StartFraction 5 plus or minus StartRoot 21 EndRoot over 2 EndFraction

D.
x equals StartFraction 5 plus or minus StartRoot 29 EndRoot over 2 EndFraction

1 answer

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:

  1. Calculate \( b^2 - 4ac \): \[ b^2 = (-5)^2 = 25 \] \[ 4ac = 4 \cdot 1 \cdot (-1) = -4 \] \[ b^2 - 4ac = 25 - (-4) = 25 + 4 = 29 \]

  2. 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. \( x = \frac{5 \pm \sqrt{29}}{2} \)