To solve the quadratic equation \( 12x^2 - 12x + 4 = -3x \), we first rearrange the equation to standard form:
\[ 12x^2 - 12x + 4 + 3x = 0 \] \[ 12x^2 - 9x + 4 = 0 \]
Next, we can use the quadratic formula, which is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our case, \( a = 12 \), \( b = -9 \), and \( c = 4 \). First, we calculate the discriminant \( b^2 - 4ac \):
\[ b^2 = (-9)^2 = 81 \] \[ 4ac = 4 \cdot 12 \cdot 4 = 192 \] \[ b^2 - 4ac = 81 - 192 = -111 \]
Since the discriminant is negative, we can conclude that the solutions will be complex. Now, substituting the values into the quadratic formula:
\[ x = \frac{-(-9) \pm \sqrt{-111}}{2(12)} = \frac{9 \pm \sqrt{-111}}{24} \]
We can write \(\sqrt{-111}\) as \(i\sqrt{111}\):
\[ x = \frac{9 \pm i\sqrt{111}}{24} \]
Now we can split the expression into its real and imaginary parts. The real part is:
\[ \frac{9}{24} = \frac{3}{8} \]
And the imaginary part is:
\[ \frac{\sqrt{111}}{24} \]
So, we can express it in the form \( a \pm bi \):
\[ x = \frac{3}{8} \pm \frac{\sqrt{111}}{24} i \]
Thus, the solutions can be expressed as:
\[ \frac{3}{8} \pm \frac{\sqrt{111}}{24} i \]
This corresponds to the second option in your provided responses:
\( \frac{3}{8} \pm \frac{\sqrt{111}}{24} i \).