Question

use the quadratic formula to solve for x.
4x^2 - 2x - 1 = 0
(if there is more than one solution, separate them with commas.)
x =

Explanation:

Step1: Identify coefficients

For the quadratic equation $4x^{2}-2x - 1=0$, we have $a = 4$, $b=-2$, $c=-1$.

Step2: Apply quadratic formula

The quadratic formula is $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. Substitute $a = 4$, $b=-2$, $c=-1$ into it:
\[

$$\begin{align*} x&=\frac{-(-2)\pm\sqrt{(-2)^{2}-4\times4\times(-1)}}{2\times4}\\ &=\frac{2\pm\sqrt{4 + 16}}{8}\\ &=\frac{2\pm\sqrt{20}}{8}\\ &=\frac{2\pm2\sqrt{5}}{8}\\ &=\frac{1\pm\sqrt{5}}{4} \end{align*}$$

\]

Answer:

$x=\frac{1+\sqrt{5}}{4},\frac{1 - \sqrt{5}}{4}$