Use the quadratic formula to solve for \( x \). \[ 6 x^{2}-6 x-1=0 \] (If there is more than one solution, separate

To solve the quadratic equation \( 6x^{2} - 6x - 1 = 0 \) using the quadratic formula, follow these steps: The quadratic formula is: \[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \] where \( a \), \( b \), and \( c \) are coefficients from the quadratic equation \( ax^{2} + bx + c = 0 \). For the given equation: - \( a = 6 \) - \( b = -6 \) - \( c = -1 \) 1. **Calculate the discriminant (\( D \))**: \[ D = b^{2} - 4ac = (-6)^{2} - 4(6)(-1) = 36 + 24 = 60 \] 2. **Compute the square root of the discriminant**: \[ \sqrt{D} = \sqrt{60} = 2\sqrt{15} \] 3. **Apply the quadratic formula**: \[ x = \frac{-(-6) \pm 2\sqrt{15}}{2 \times 6} = \frac{6 \pm 2\sqrt{15}}{12} \] 4. **Simplify the expression**: \[ x = \frac{6 \pm 2\sqrt{15}}{12} = \frac{3 \pm \sqrt{15}}{6} \] **Solutions**: \[ x = \frac{3 + \sqrt{15}}{6}, \quad x = \frac{3 - \sqrt{15}}{6} \] **Final Answer**: \[ x = \frac{3 + \sqrt{15}}{6}, \quad \frac{3 - \sqrt{15}}{6} \]