Solve the following equation using the quadratic formula. \( -6 x^{2}=-3 x+14 \)

To solve the equation \( -6 x^{2} = -3 x + 14 \) using the quadratic formula, we first need to rearrange it into the standard form \( ax^2 + bx + c = 0 \). 1. **Rearranging the equation:** \[ -6x^2 + 3x - 14 = 0 \] This gives us: - \( a = -6 \) - \( b = 3 \) - \( c = -14 \) 2. **Using the quadratic formula:** The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] 3. **Calculating the discriminant:** First, we calculate \( b^2 - 4ac \): \[ b^2 - 4ac = 3^2 - 4(-6)(-14) \] Now, let's calculate this step by step. - Calculate \( 3^2 \): \[ 3^2 = 9 \] - Calculate \( 4 \times -6 \times -14 \): \[ 4 \times -6 \times -14 = 336 \] - Now, substitute these values into the discriminant: \[ 9 - 336 = -327 \] Since the discriminant is negative, this indicates that the equation has no real solutions, but two complex solutions. 4. **Calculating the solutions:** We will now calculate the complex solutions using the quadratic formula: \[ x = \frac{-3 \pm \sqrt{-327}}{2 \times -6} \] Let's calculate this. - The square root of \(-327\) can be expressed as: \[ \sqrt{-327} = i\sqrt{327} \] Now, we can substitute this back into the formula: \[ x = \frac{-3 \pm i\sqrt{327}}{-12} \] This simplifies to: \[ x = \frac{3}{12} \mp \frac{i\sqrt{327}}{12} \] \[ x = \frac{1}{4} \mp \frac{i\sqrt{327}}{12} \] Thus, the solutions to the equation \( -6 x^{2} = -3 x + 14 \) are: \[ x = \frac{1}{4} + \frac{i\sqrt{327}}{12} \quad \text{and} \quad x = \frac{1}{4} - \frac{i\sqrt{327}}{12} \]