Question
Djamila is looking for conjugate pair solutions to the equation 3x2−5x=−6 . Which of the following options should she choose?(1 point)
Answers
GPT-4o mini
To find conjugate pair solutions for the equation \( 3x^2 - 5x = -6 \), we first rewrite the equation in standard form:
\[
3x^2 - 5x + 6 = 0
\]
Next, we can check if the solutions are conjugate pairs by calculating the discriminant \( D \), which is given by the formula:
\[
D = b^2 - 4ac
\]
In our quadratic equation, \( a = 3 \), \( b = -5 \), and \( c = 6 \).
Let's calculate the discriminant:
\[
D = (-5)^2 - 4(3)(6) = 25 - 72 = -47
\]
Since the discriminant is negative (\( D < 0 \)), this means that the quadratic equation has two complex conjugate solutions.
To find the solutions, we use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{D}}{2a}
\]
Inserting the values:
\[
x = \frac{-(-5) \pm \sqrt{-47}}{2(3)} = \frac{5 \pm i\sqrt{47}}{6}
\]
Thus, the solutions are:
\[
x_1 = \frac{5 + i\sqrt{47}}{6} \quad \text{and} \quad x_2 = \frac{5 - i\sqrt{47}}{6}
\]
These solutions are indeed a conjugate pair.
Based on this analysis, Djamila should choose the option that indicates the existence of complex conjugate solutions. If you need specific options to choose from, please provide them!
\[
3x^2 - 5x + 6 = 0
\]
Next, we can check if the solutions are conjugate pairs by calculating the discriminant \( D \), which is given by the formula:
\[
D = b^2 - 4ac
\]
In our quadratic equation, \( a = 3 \), \( b = -5 \), and \( c = 6 \).
Let's calculate the discriminant:
\[
D = (-5)^2 - 4(3)(6) = 25 - 72 = -47
\]
Since the discriminant is negative (\( D < 0 \)), this means that the quadratic equation has two complex conjugate solutions.
To find the solutions, we use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{D}}{2a}
\]
Inserting the values:
\[
x = \frac{-(-5) \pm \sqrt{-47}}{2(3)} = \frac{5 \pm i\sqrt{47}}{6}
\]
Thus, the solutions are:
\[
x_1 = \frac{5 + i\sqrt{47}}{6} \quad \text{and} \quad x_2 = \frac{5 - i\sqrt{47}}{6}
\]
These solutions are indeed a conjugate pair.
Based on this analysis, Djamila should choose the option that indicates the existence of complex conjugate solutions. If you need specific options to choose from, please provide them!