Solve the quadratic equation 6x2−3x+6=0

To solve the quadratic equation \( 6x^2 - 3x + 6 = 0 \), we can use the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

In this equation:

• \( a = 6 \)
• \( b = -3 \)
• \( c = 6 \)

Calculating the discriminant (\( b^2 - 4ac \)):

\[ b^2 = (-3)^2 = 9 \] \[ 4ac = 4 \times 6 \times 6 = 144 \] \[ b^2 - 4ac = 9 - 144 = -135 \]

Since the discriminant is negative, the solutions will be complex. Now we can substitute into the quadratic formula:

\[ x = \frac{-(-3) \pm \sqrt{-135}}{2 \times 6} \] \[ x = \frac{3 \pm \sqrt{-135}}{12} \]

To express \( \sqrt{-135} \) in terms of \( i \):

\[ \sqrt{-135} = \sqrt{135}i = \sqrt{9 \cdot 15}i = 3\sqrt{15}i \]

Now substitute back into the equation:

\[ x = \frac{3 \pm 3\sqrt{15}i}{12} \] \[ x = \frac{3}{12} \pm \frac{3\sqrt{15}}{12}i \] \[ x = \frac{1}{4} \pm \frac{\sqrt{15}}{4}i \]

Thus, the solutions can be expressed as:

\[ x = \frac{1}{4} \pm \frac{\sqrt{15}}{4}i \]

From the options provided, the correct response is:

\[ \frac{1}{4} \pm \frac{\sqrt{15}}{4}i \]