Question
Djamila is looking for conjugate pair solutions to the equation 3x2−5x=−6. Which of the following options should she choose?(1 point)%0D%0AResponses%0D%0A%0D%0A56±i47√6%0D%0AStart Fraction 5 over 6 End Fraction plus or minus Start Fraction i Start Root 47 End Root over 6 End Fraction%0D%0A%0D%0A56±5i6%0D%0AStart Fraction 5 over 6 End Fraction plus or minus Start Fraction 5 i over 6 End Fraction%0D%0A%0D%0A56±7i6%0D%0AStart Fraction 5 over 6 End Fraction plus or minus Start Fraction 7 i over 6 End Fraction%0D%0A%0D%0A−56±i47√6
Answers
GPT-4o mini
To find the conjugate pair solutions to the equation \(3x^2 - 5x = -6\), we first rewrite it in standard form:
\[
3x^2 - 5x + 6 = 0.
\]
Next, we can use the quadratic formula to find the solutions:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
\]
Here, \(a = 3\), \(b = -5\), and \(c = 6\). We first calculate the discriminant:
\[
b^2 - 4ac = (-5)^2 - 4 \cdot 3 \cdot 6 = 25 - 72 = -47.
\]
Since the discriminant is negative, this indicates that the solutions will be complex numbers (conjugate pairs).
Substituting the values into the quadratic formula gives:
\[
x = \frac{-(-5) \pm \sqrt{-47}}{2 \cdot 3} = \frac{5 \pm i\sqrt{47}}{6}.
\]
Thus, the solutions to the equation are:
\[
x = \frac{5}{6} \pm \frac{i\sqrt{47}}{6}.
\]
From the provided options, the correct choice is:
**Start Fraction 5 over 6 End Fraction plus or minus Start Fraction i Start Root 47 End Root over 6 End Fraction.**
\[
3x^2 - 5x + 6 = 0.
\]
Next, we can use the quadratic formula to find the solutions:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
\]
Here, \(a = 3\), \(b = -5\), and \(c = 6\). We first calculate the discriminant:
\[
b^2 - 4ac = (-5)^2 - 4 \cdot 3 \cdot 6 = 25 - 72 = -47.
\]
Since the discriminant is negative, this indicates that the solutions will be complex numbers (conjugate pairs).
Substituting the values into the quadratic formula gives:
\[
x = \frac{-(-5) \pm \sqrt{-47}}{2 \cdot 3} = \frac{5 \pm i\sqrt{47}}{6}.
\]
Thus, the solutions to the equation are:
\[
x = \frac{5}{6} \pm \frac{i\sqrt{47}}{6}.
\]
From the provided options, the correct choice is:
**Start Fraction 5 over 6 End Fraction plus or minus Start Fraction i Start Root 47 End Root over 6 End Fraction.**