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%0A−56±i47√6%0D%0Anegative Start 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
Answers
GPT-4o mini
To solve the equation \( 3x^2 - 5x = -6 \), we first rearrange it to standard form:
\[
3x^2 - 5x + 6 = 0
\]
Next, we can use the quadratic formula to find the roots of the equation \( ax^2 + bx + c = 0 \), which is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
In our case, \( a = 3 \), \( b = -5 \), and \( c = 6 \). Now we will calculate the discriminant:
\[
b^2 - 4ac = (-5)^2 - 4 \cdot 3 \cdot 6 = 25 - 72 = -47
\]
Since the discriminant is negative (\(-47\)), we will have complex (conjugate pair) solutions. Now let's apply the quadratic formula:
\[
x = \frac{-(-5) \pm \sqrt{-47}}{2 \cdot 3} = \frac{5 \pm \sqrt{-47}}{6}
\]
This can be rewritten as:
\[
x = \frac{5}{6} \pm \frac{i \sqrt{47}}{6}
\]
Now we can match our results with the options provided. The conjugate pair solutions are:
\[
\frac{5}{6} \pm \frac{i \sqrt{47}}{6}
\]
Thus, the correct option that matches this solution format 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 roots of the equation \( ax^2 + bx + c = 0 \), which is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
In our case, \( a = 3 \), \( b = -5 \), and \( c = 6 \). Now we will calculate the discriminant:
\[
b^2 - 4ac = (-5)^2 - 4 \cdot 3 \cdot 6 = 25 - 72 = -47
\]
Since the discriminant is negative (\(-47\)), we will have complex (conjugate pair) solutions. Now let's apply the quadratic formula:
\[
x = \frac{-(-5) \pm \sqrt{-47}}{2 \cdot 3} = \frac{5 \pm \sqrt{-47}}{6}
\]
This can be rewritten as:
\[
x = \frac{5}{6} \pm \frac{i \sqrt{47}}{6}
\]
Now we can match our results with the options provided. The conjugate pair solutions are:
\[
\frac{5}{6} \pm \frac{i \sqrt{47}}{6}
\]
Thus, the correct option that matches this solution format is:
**Start Fraction 5 over 6 End Fraction plus or minus Start Fraction i Start Root 47 End Root over 6 End Fraction**.