Question

Djamila is looking for conjugate pair solutions to the equation 3x2−5x=−6
3
𝑥
2
−
5
𝑥
=
−
6
. Which of the following options should she choose?(1 point)
Responses

56±5i6
5
6
±
5
𝑖
6
Start Fraction 5 over 6 End Fraction plus or minus Start Fraction 5 i over 6 End Fraction

−56±i47√6
−
5
6
±
𝑖
47
6
negative Start Fraction 5 over 6 End Fraction plus or minus Start Fraction i Start Root 47 End Root over 6 End Fraction

56±7i6
5
6
±
7
𝑖
6
Start Fraction 5 over 6 End Fraction plus or minus Start Fraction 7 i over 6 End Fraction

56±i47√6

Answer 1

To solve the equation \(3x^2 - 5x = -6\), we can rearrange it to standard quadratic form:

\[
3x^2 - 5x + 6 = 0
\]

Next, we can apply the quadratic formula, given by:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

where \(a = 3\), \(b = -5\), and \(c = 6\).

Calculating 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 be complex conjugates. Now substituting into the quadratic formula:

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

This can be simplified to:

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

Now we can match the calculated solutions to the options provided. The correct option is:

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

Thus, Djamila should choose:

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