To solve the equation \( 3x^2 - 5x = -6 \), let's first rewrite it in standard form:
\[ 3x^2 - 5x + 6 = 0 \]
Next, we'll apply the quadratic formula, which is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In this equation:
- \( a = 3 \)
- \( b = -5 \)
- \( c = 6 \)
Now, we can calculate the discriminant \( b^2 - 4ac \):
\[ b^2 = (-5)^2 = 25 \] \[ 4ac = 4 \cdot 3 \cdot 6 = 72 \] \[ b^2 - 4ac = 25 - 72 = -47 \]
Since the discriminant is negative (\(-47\)), the solutions will be complex conjugates. Now we can plug our values into the quadratic formula:
\[ x = \frac{-(-5) \pm \sqrt{-47}}{2 \cdot 3} = \frac{5 \pm \sqrt{-47}}{6} \]
This simplifies further:
\[ x = \frac{5 \pm i \sqrt{47}}{6} \]
Based on this, Djamila should choose:
\[ \frac{5}{6} \pm \frac{i \sqrt{47}}{6} \]
From the given options, this matches:
\[ \frac{5}{6} \pm \frac{i \sqrt{47}}{6} \]
So the correct choice is:
\( \frac{5}{6} \pm \frac{i \sqrt{47}}{6} \).