Solve using the quadratic formula. \[ 6 w^{2}+w-6=0 \]

To solve the quadratic equation \( 6w^{2} + w - 6 = 0 \) using the quadratic formula, we identify the coefficients: \( a = 6 \), \( b = 1 \), and \( c = -6 \). The quadratic formula is given by: \[ w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Substituting in the values, we compute the discriminant \( b^2 - 4ac \): \[ 1^2 - 4(6)(-6) = 1 + 144 = 145 \] Now we can plug this into the formula: \[ w = \frac{-1 \pm \sqrt{145}}{2 \cdot 6} = \frac{-1 \pm \sqrt{145}}{12} \] Thus, the solutions for \( w \) are: \[ w = \frac{-1 + \sqrt{145}}{12} \quad \text{and} \quad w = \frac{-1 - \sqrt{145}}{12} \]