To solve the equation \( 3x^2 - 5x + 6 = 0 \) (which is rearranged from \( 3x^2 - 5x = -6 \)), we will use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 3 \), \( b = -5 \), and \( c = 6 \). Let's calculate the discriminant \( b^2 - 4ac \):
\[ b^2 = (-5)^2 = 25 \] \[ 4ac = 4 \cdot 3 \cdot 6 = 72 \] Thus, the discriminant is:
\[ b^2 - 4ac = 25 - 72 = -47 \]
Since the discriminant is negative, we will have complex (conjugate pair) solutions. Now we can substitute the values into the quadratic formula:
\[ x = \frac{-(-5) \pm \sqrt{-47}}{2 \cdot 3} = \frac{5 \pm \sqrt{47}i}{6} \]
This means the solutions are:
\[ x = \frac{5}{6} \pm \frac{\sqrt{47}}{6}i \]
Now, we can investigate the options provided:
- \( \frac{5}{6} \pm \frac{7i}{6} \) - This does not match.
- \( \frac{5}{6} \pm \frac{i \sqrt{47}}{6} \) - This matches our solution.
- \( -\frac{5}{6} \pm \frac{i \sqrt{47}}{6} \) - This does not match.
- \( \frac{5}{6} \pm \frac{5i}{6} \) - This does not match.
The correct option is:
Start Fraction 5 over 6 End Fraction plus or minus Start Fraction i Start Root 47 End Root over 6 End Fraction.