To find the conjugate pair solutions to the equation \(3x^2 - 5x = -6\), we first rearrange the equation into standard quadratic form:
\[ 3x^2 - 5x + 6 = 0. \]
Next, we can use the quadratic formula to find the roots of the equation, where \(a = 3\), \(b = -5\), and \(c = 6\). The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \]
Let's calculate the discriminant \(D = b^2 - 4ac\):
\[ D = (-5)^2 - 4 \cdot 3 \cdot 6 = 25 - 72 = -47. \]
Since the discriminant is negative (\(D < 0\)), the solutions will be complex (conjugate pair solutions).
Now, we can compute the roots using the quadratic formula:
\[ x = \frac{5 \pm \sqrt{-47}}{2 \cdot 3} = \frac{5 \pm i\sqrt{47}}{6}. \]
This gives us the conjugate pair solutions:
\[ x_1 = \frac{5 + i\sqrt{47}}{6}, \quad x_2 = \frac{5 - i\sqrt{47}}{6}. \]
If options were provided, Djamila should choose the one that corresponds to these conjugate pair solutions. If you have specific options to consider, please let me know!
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