Question

Complex Numbers & Higher Order Polynomials Quick Check
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Question
Djamila is looking for conjugate pair solutions to the equation 3x2−5x=−6
. Which of the following options should she choose?(1 point)
Responses

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

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

56±5i6

Answer 1

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.