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 Start Fraction 5 over 6 End Fraction plus or minus Start Fraction i Start Root 47 End Root over 6 End Fraction 56±5i6 Start Fraction 5 over 6 End Fraction plus or minus Start Fraction 5 i over 6 End Fraction −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±7i6
Answers
GPT-4o mini
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 can use the quadratic formula to find the solutions:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \(a = 3\), \(b = -5\), and \(c = 6\). Let's calculate the discriminant \(D = b^2 - 4ac\):
\[
D = (-5)^2 - 4 \cdot 3 \cdot 6 = 25 - 72 = -47
\]
Since the discriminant is negative (\(-47\)), the solutions will be complex conjugates.
Now, substituting back into the quadratic formula:
\[
x = \frac{-(-5) \pm \sqrt{-47}}{2 \cdot 3} = \frac{5 \pm i\sqrt{47}}{6}
\]
Thus, the solutions are:
\[
x = \frac{5}{6} \pm \frac{i\sqrt{47}}{6}
\]
Looking at the options provided:
- \(56 \pm i\frac{\sqrt{47}}{6}\) \( \Rightarrow \) This option is not valid since the \(5\) is not in the numerator.
- \(\frac{5}{6} \pm \frac{i\sqrt{47}}{6}\) \( \Rightarrow\) This matches our solution.
- \(56 \pm \frac{5i}{6}\) \( \Rightarrow \) Incorrect pair since both have the wrong terms.
- \(-\frac{5}{6} \pm \frac{i\sqrt{47}}{6}\) \( \Rightarrow \) This option is also incorrect.
- \(56 \pm 7i\) \( \Rightarrow \) Invalid since the terms do not match.
Therefore, the correct answer is:
\(\frac{5}{6} \pm \frac{i\sqrt{47}}{6}\)
\[
3x^2 - 5x + 6 = 0
\]
Next, we can use the quadratic formula to find the solutions:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \(a = 3\), \(b = -5\), and \(c = 6\). Let's calculate the discriminant \(D = b^2 - 4ac\):
\[
D = (-5)^2 - 4 \cdot 3 \cdot 6 = 25 - 72 = -47
\]
Since the discriminant is negative (\(-47\)), the solutions will be complex conjugates.
Now, substituting back into the quadratic formula:
\[
x = \frac{-(-5) \pm \sqrt{-47}}{2 \cdot 3} = \frac{5 \pm i\sqrt{47}}{6}
\]
Thus, the solutions are:
\[
x = \frac{5}{6} \pm \frac{i\sqrt{47}}{6}
\]
Looking at the options provided:
- \(56 \pm i\frac{\sqrt{47}}{6}\) \( \Rightarrow \) This option is not valid since the \(5\) is not in the numerator.
- \(\frac{5}{6} \pm \frac{i\sqrt{47}}{6}\) \( \Rightarrow\) This matches our solution.
- \(56 \pm \frac{5i}{6}\) \( \Rightarrow \) Incorrect pair since both have the wrong terms.
- \(-\frac{5}{6} \pm \frac{i\sqrt{47}}{6}\) \( \Rightarrow \) This option is also incorrect.
- \(56 \pm 7i\) \( \Rightarrow \) Invalid since the terms do not match.
Therefore, the correct answer is:
\(\frac{5}{6} \pm \frac{i\sqrt{47}}{6}\)