Djamila is looking for conjugate pair solutions to the equation 3x2−5x=−6. Which of the following options should she choose?(1 point)Responses56±7i6Start Fraction 5 over 6 End Fraction plus or minus Start Fraction 7 i over 6 End Fraction56±5i6Start Fraction 5 over 6 End Fraction plus or minus Start Fraction 5 i over 6 End Fraction56±i47√6Start Fraction 5 over 6 End Fraction plus or minus Start Fraction i Start Root 47 End Root over 6 End Fraction−56±i47√6

Answer 1

To solve the equation \(3x^2 - 5x = -6\), we first rearrange it into standard quadratic 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 \(c = 6\). First, we calculate the discriminant \(b^2 - 4ac\):

\[ b^2 = (-5)^2 = 25 \] \[ 4ac = 4 \cdot 3 \cdot 6 = 72 \] \[ b^2 - 4ac = 25 - 72 = -47 \]

Since the discriminant is negative, the solutions will involve imaginary numbers. We can now substitute back into the quadratic formula:

\[ x = \frac{-(-5) \pm \sqrt{-47}}{2 \cdot 3} = \frac{5 \pm \sqrt{-47}}{6} \]

This can be simplified to:

\[ x = \frac{5}{6} \pm \frac{i \sqrt{47}}{6} \]

Thus, the conjugate pair of solutions is:

\[ \frac{5}{6} \pm \frac{i \sqrt{47}}{6} \]

Looking at the given options, the correct choice matching our solutions is:

\[ \frac{5}{6} \pm \frac{i \sqrt{47}}{6} \]

Therefore, Djamila should choose:

−56±i47√6.