Solve for \( t \) \[ 6 t^{2}+7 t+2=0 \]

We start with the equation \[ 6t^2 + 7t + 2 = 0. \] **Step 1: Identify the coefficients** The quadratic equation has the standard form \[ at^2 + bt + c = 0, \] with \[ a = 6, \quad b = 7, \quad c = 2. \] **Step 2: Compute the discriminant** The discriminant is given by \[ \Delta = b^2 - 4ac. \] Substitute the values: \[ \Delta = 7^2 - 4 \cdot 6 \cdot 2 = 49 - 48 = 1. \] **Step 3: Apply the quadratic formula** The quadratic formula is \[ t = \frac{-b \pm \sqrt{\Delta}}{2a}. \] Substitute the values: \[ t = \frac{-7 \pm \sqrt{1}}{2 \cdot 6} = \frac{-7 \pm 1}{12}. \] **Step 4: Find the two solutions** 1. For the positive square root: \[ t = \frac{-7 + 1}{12} = \frac{-6}{12} = -\frac{1}{2}. \] 2. For the negative square root: \[ t = \frac{-7 - 1}{12} = \frac{-8}{12} = -\frac{2}{3}. \] **Final Answer:** \[ t = -\frac{1}{2} \quad \text{or} \quad t = -\frac{2}{3}. \]