2х^2-х-1=0

To solve the quadratic equation \(2x^2 - x - 1 = 0\), we can use the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

where \(a = 2\), \(b = -1\), and \(c = -1\).

First, we calculate the discriminant:

\[ b^2 - 4ac = (-1)^2 - 4 \cdot 2 \cdot (-1) = 1 + 8 = 9 \]

Now, we can plug the values into the quadratic formula:

\[ x = \frac{-(-1) \pm \sqrt{9}}{2 \cdot 2} \] \[ x = \frac{1 \pm 3}{4} \]

This gives us two solutions:

1. \[ x = \frac{1 + 3}{4} = \frac{4}{4} = 1 \]

2. \[ x = \frac{1 - 3}{4} = \frac{-2}{4} = -\frac{1}{2} \]

Thus, the solutions to the equation \(2x^2 - x - 1 = 0\) are:

\[ x = 1 \quad \text{and} \quad x = -\frac{1}{2} \]