Solve the quadratic equ -8 \( x^{2}-6 x-2=0 \)

To solve the quadratic equation \(x^{2}-6x-2=0\), we can use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] In this equation, \(a = 1\), \(b = -6\), and \(c = -2\). Plugging these values into the formula gives us: \[ x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4 \cdot 1 \cdot (-2)}}{2 \cdot 1} \] Calculating this step-by-step, we first find: \[ b^2 - 4ac = 36 + 8 = 44 \] So the formula becomes: \[ x = \frac{6 \pm \sqrt{44}}{2} \] Now, simplify \(\sqrt{44}\) to \(2\sqrt{11}\): \[ x = \frac{6 \pm 2\sqrt{11}}{2} \] Dividing everything by 2, we get: \[ x = 3 \pm \sqrt{11} \] Thus, the solutions are: \[ x = 3 + \sqrt{11} \quad \text{and} \quad x = 3 - \sqrt{11} \]