To solve the equation \(3x^2 - 5x = -6\), we first rewrite it in standard form:
\[ 3x^2 - 5x + 6 = 0. \]
Next, we can apply the quadratic formula, which is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \]
where \(a = 3\), \(b = -5\), and \(c = 6\).
Now, we can calculate the discriminant \(b^2 - 4ac\):
\[ b^2 - 4ac = (-5)^2 - 4 \cdot 3 \cdot 6 = 25 - 72 = -47. \]
Since the discriminant is negative, this indicates that there are complex (imaginary) solutions. Now we can substitute the values into the quadratic formula:
\[ x = \frac{-(-5) \pm \sqrt{-47}}{2 \cdot 3} = \frac{5 \pm \sqrt{-47}}{6}. \]
Now we can express \(\sqrt{-47}\) in terms of \(i\):
\[ \sqrt{-47} = i\sqrt{47}. \]
Thus, we can rewrite the solution as:
\[ x = \frac{5 \pm i\sqrt{47}}{6}. \]
Therefore, the conjugate pair solutions to the equation are:
\[ \frac{5}{6} \pm \frac{i\sqrt{47}}{6}. \]
Based on the provided options, the correct choice for the solutions 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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