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