To solve the equation \( 3x^2 - 5x = -6 \), we first rearrange it to standard form:
\[ 3x^2 - 5x + 6 = 0 \]
Next, we can use the quadratic formula to find the roots of the equation \( ax^2 + bx + c = 0 \), which is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our case, \( a = 3 \), \( b = -5 \), and \( c = 6 \). Now we will calculate the discriminant:
\[ b^2 - 4ac = (-5)^2 - 4 \cdot 3 \cdot 6 = 25 - 72 = -47 \]
Since the discriminant is negative (\(-47\)), we will have complex (conjugate pair) solutions. Now let's apply the quadratic formula:
\[ x = \frac{-(-5) \pm \sqrt{-47}}{2 \cdot 3} = \frac{5 \pm \sqrt{-47}}{6} \]
This can be rewritten as:
\[ x = \frac{5}{6} \pm \frac{i \sqrt{47}}{6} \]
Now we can match our results with the options provided. The conjugate pair solutions are:
\[ \frac{5}{6} \pm \frac{i \sqrt{47}}{6} \]
Thus, the correct option that matches this solution format is:
Start Fraction 5 over 6 End Fraction plus or minus Start Fraction i Start Root 47 End Root over 6 End Fraction.
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