To find the conjugate pair solutions to the equation \(3x^2 - 5x = -6\), we first rewrite it in standard form:
\[ 3x^2 - 5x + 6 = 0. \]
Next, we can use the quadratic formula to find the solutions:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \]
Here, \(a = 3\), \(b = -5\), and \(c = 6\). We first calculate the discriminant:
\[ b^2 - 4ac = (-5)^2 - 4 \cdot 3 \cdot 6 = 25 - 72 = -47. \]
Since the discriminant is negative, this indicates that the solutions will be complex numbers (conjugate pairs).
Substituting the values into the quadratic formula gives:
\[ x = \frac{-(-5) \pm \sqrt{-47}}{2 \cdot 3} = \frac{5 \pm i\sqrt{47}}{6}. \]
Thus, the solutions to the equation are:
\[ x = \frac{5}{6} \pm \frac{i\sqrt{47}}{6}. \]
From the provided options, the correct choice is:
Start Fraction 5 over 6 End Fraction plus or minus Start Fraction i Start Root 47 End Root over 6 End Fraction.