To solve the quadratic equation \( 6x^2 - 3x + 6 = 0 \), we will use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 6 \), \( b = -3 \), and \( c = 6 \).
- Calculate the discriminant \( b^2 - 4ac \):
\[ b^2 - 4ac = (-3)^2 - 4(6)(6) = 9 - 144 = -135 \]
- Since the discriminant is negative, we will have complex solutions. Now we compute \( \sqrt{-135} \):
\[ \sqrt{-135} = \sqrt{135} \cdot i = \sqrt{9 \cdot 15} \cdot i = 3\sqrt{15} \cdot i \]
- Now substitute back into the quadratic formula:
\[ x = \frac{-(-3) \pm 3\sqrt{15} i}{2 \cdot 6} = \frac{3 \pm 3\sqrt{15} i}{12} = \frac{1}{4} \pm \frac{\sqrt{15}}{4} i \]
Thus, the solutions for the equation \( 6x^2 - 3x + 6 = 0 \) are:
\[ x = \frac{1}{4} \pm \frac{\sqrt{15}}{4} i \]
The correct option that expresses its solutions in the form \( a \pm bi \) is:
\[ \boxed{\frac{1}{4} \pm \frac{\sqrt{15}}{4} i} \]