To find the conjugate pair solutions to the equation \(3x^2 - 5x = -6\), we first rewrite the equation in standard form:
\[ 3x^2 - 5x + 6 = 0. \]
Next, we can apply the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 3\), \(b = -5\), and \(c = 6\).
Calculating the discriminant:
\[ b^2 - 4ac = (-5)^2 - 4(3)(6) = 25 - 72 = -47. \]
Since the discriminant is negative, the solutions will indeed be complex (conjugate pair solutions).
Now applying the quadratic formula:
\[ x = \frac{-(-5) \pm \sqrt{-47}}{2(3)} = \frac{5 \pm \sqrt{-47}}{6} = \frac{5 \pm i\sqrt{47}}{6}. \]
Thus the solutions are:
\[ x = \frac{5}{6} \pm \frac{\sqrt{47}}{6}i. \]
Among the provided options, this matches:
Start Fraction 5 over 6 End Fraction plus or minus Start Fraction i Start Root 47 End Root over 6 End Fraction.
So, Djamila should choose this option:
\[ \frac{5}{6} \pm \frac{i\sqrt{47}}{6}. \]
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