Question

what are the solutions to the equation shown?
x² - 5x - 1 = 0
a. x = (5 ± √6)/2
b. x = (5 ± √14)/2
c. x = (5 ± √21)/2
d. x = (5 ± √29)/2

Explanation:

Step1: Identify the quadratic formula

The quadratic formula for a quadratic equation \(ax^2 + bx + c = 0\) is \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\).

Step2: Determine \(a\), \(b\), and \(c\)

For the equation \(x^2 - 5x - 1 = 0\), we have \(a = 1\), \(b=-5\), and \(c = -1\).

Step3: Calculate the discriminant \(b^2 - 4ac\)

Substitute the values of \(a\), \(b\), and \(c\) into the discriminant formula:

$$
LATEXBLOCK0
$$

Step4: Apply the quadratic formula

Substitute \(a = 1\), \(b=-5\), and the discriminant (29) into the quadratic formula:

$$ x=\frac{-(-5)\pm\sqrt{29}}{2\times1}=\frac{5\pm\sqrt{29}}{2} $$

Answer:

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