A. \( 3 x^{2}-x-10 \)

To solve the quadratic equation \( 3x^{2} - x - 10 = 0 \), we can use the quadratic formula, which is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a \), \( b \), and \( c \) are the coefficients from the equation \( ax^2 + bx + c = 0 \). In our case: - \( a = 3 \) - \( b = -1 \) - \( c = -10 \) Now, let's calculate the discriminant \( b^2 - 4ac \) first: \[ b^2 - 4ac = (-1)^2 - 4 \cdot 3 \cdot (-10) \] Calculating this step by step: 1. Calculate \( (-1)^2 \): \[ 1 \] 2. Calculate \( 4 \cdot 3 \cdot (-10) \): \[ 4 \cdot 3 = 12 \] \[ 12 \cdot (-10) = -120 \] 3. Now, combine the results: \[ 1 - (-120) = 1 + 120 = 121 \] Now that we have the discriminant, we can substitute it back into the quadratic formula: \[ x = \frac{-(-1) \pm \sqrt{121}}{2 \cdot 3} \] Calculating this step by step: 1. Calculate \( -(-1) \): \[ 1 \] 2. Calculate \( \sqrt{121} \): \[ 11 \] 3. Now substitute these values into the formula: \[ x = \frac{1 \pm 11}{6} \] This gives us two possible solutions: 1. For \( x = \frac{1 + 11}{6} \): \[ x = \frac{12}{6} = 2 \] 2. For \( x = \frac{1 - 11}{6} \): \[ x = \frac{-10}{6} = -\frac{5}{3} \] Thus, the solutions to the equation \( 3x^{2} - x - 10 = 0 \) are: \[ x = 2 \quad \text{and} \quad x = -\frac{5}{3} \]