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 will use the quadratic formula to find the solutions. The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \]
where \(a = 3\), \(b = -5\), and \(c = 6\).
First, we calculate the discriminant \(b^2 - 4ac\):
\[ b^2 - 4ac = (-5)^2 - 4(3)(6) = 25 - 72 = -47. \]
Since the discriminant is negative, the solutions will be complex conjugates.
Now, substituting into the quadratic formula:
\[ x = \frac{-(-5) \pm \sqrt{-47}}{2(3)} = \frac{5 \pm i\sqrt{47}}{6}. \]
Thus, the solutions can be expressed as:
\[ x = \frac{5}{6} \pm \frac{i\sqrt{47}}{6}. \]
Among the given options, Djamila should choose:
\[ \text{negative } \frac{5}{6} \pm \frac{i\sqrt{47}}{6}. \]
This matches the form that we derived.
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