To solve the quadratic equation \(12x^2 - 12x + 4 = -3x\), we first rearrange the equation:
\[ 12x^2 - 12x + 4 + 3x = 0 \] \[ 12x^2 - 9x + 4 = 0 \]
Now we can use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 12\), \(b = -9\), and \(c = 4\).
First, we need to calculate the discriminant:
\[ b^2 - 4ac = (-9)^2 - 4(12)(4) \] \[ = 81 - 192 \] \[ = -111 \]
Since the discriminant is negative, the solutions will be complex. Next, we substitute into the quadratic formula:
\[ x = \frac{-(-9) \pm \sqrt{-111}}{2(12)} \] \[ = \frac{9 \pm \sqrt{-111}}{24} \]
We can express \(\sqrt{-111}\) as \(i\sqrt{111}\):
\[ x = \frac{9 \pm i\sqrt{111}}{24} \]
Now we can split this into real and imaginary parts:
\[ x = \frac{9}{24} \pm \frac{\sqrt{111}}{24}i \] \[ = \frac{3}{8} \pm \frac{\sqrt{111}}{24}i \]
Thus, the solutions in the required form \(a \pm bi\) are:
\[ \frac{3}{8} \pm \frac{\sqrt{111}}{24}i \]
Now, looking at your options, we see that this corresponds to:
Start Fraction 3 over 8 End Fraction plus or minus Start Fraction Start Root 111 End Root over 24 End Fraction i.