Questions: Use the quadratic formula to solve for x. 9x^2-3x-1=0 (If there is more than one solution, separate them with commas.) x=

Transcript text: Use the quadratic formula to solve for $x$. \[ 9 x^{2}-3 x-1=0 \] (If there is more than one solution, separate them with commas.) \[ x= \]

Solution

Solution Steps

Step 1: Identify Coefficients

The given quadratic equation is $9x^{2} - 3x - 1 = 0$. Here, we identify the coefficients as follows:

$a = 9$
$b = -3$
$c = -1$

Step 2: Calculate the Discriminant

We calculate the discriminant using the formula $D = b^2 - 4ac$: \[ D = (-3)^2 - 4 \cdot 9 \cdot (-1) = 9 + 36 = 45 \]

Step 3: Apply the Quadratic Formula

Using the quadratic formula $x = \frac{-b \pm \sqrt{D}}{2a}$, we find the two solutions: \[ x_1 = \frac{-(-3) + \sqrt{45}}{2 \cdot 9} = \frac{3 + 3\sqrt{5}}{18} = \frac{1 + \sqrt{5}}{6} \] \[ x_2 = \frac{-(-3) - \sqrt{45}}{2 \cdot 9} = \frac{3 - 3\sqrt{5}}{18} = \frac{1 - \sqrt{5}}{6} \]