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**, which is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] For the given equation, the coefficients are: - \( a = 1 \) - \( b = -6 \) - \( c = -2 \) **Step 1: Calculate the Discriminant (\( \Delta \))** \[ \Delta = b^2 - 4ac = (-6)^2 - 4(1)(-2) = 36 + 8 = 44 \] **Step 2: Apply the Quadratic Formula** \[ x = \frac{-(-6) \pm \sqrt{44}}{2 \times 1} = \frac{6 \pm \sqrt{44}}{2} \] Simplify \( \sqrt{44} \): \[ \sqrt{44} = \sqrt{4 \times 11} = 2\sqrt{11} \] So, \[ x = \frac{6 \pm 2\sqrt{11}}{2} = 3 \pm \sqrt{11} \] **Solutions:** \[ x = 3 + \sqrt{11} \quad \text{and} \quad x = 3 - \sqrt{11} \] **Approximate Values:** - \( \sqrt{11} \approx 3.3166 \) - \( x \approx 3 + 3.3166 = 6.3166 \) - \( x \approx 3 - 3.3166 = -0.3166 \) **Final Answer:** \[ x = 3 + \sqrt{11} \quad \text{and} \quad x = 3 - \sqrt{11} \]