To solve the quadratic equation [tex]\(3x^2 - 2x + 4 = 0\)[/tex], we will use the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Here, the coefficients are [tex]\(a = 3\)[/tex], [tex]\(b = -2\)[/tex], and [tex]\(c = 4\)[/tex].
1. Calculate the discriminant:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
[tex]\[
\Delta = (-2)^2 - 4 \cdot 3 \cdot 4
\][/tex]
[tex]\[
\Delta = 4 - 48
\][/tex]
[tex]\[
\Delta = -44
\][/tex]
Since the discriminant [tex]\(\Delta\)[/tex] is negative, this indicates that the roots are complex.
2. Find the roots using the quadratic formula:
[tex]\[
x = \frac{-(-2) \pm \sqrt{-44}}{2 \cdot 3}
\][/tex]
[tex]\[
x = \frac{2 \pm \sqrt{-44}}{6}
\][/tex]
We know that [tex]\(\sqrt{-44} = \sqrt{44} \cdot i\)[/tex].
[tex]\[
x = \frac{2 \pm \sqrt{44} \cdot i}{6}
\][/tex]
Simplify the square root of 44:
[tex]\[
\sqrt{44} = \sqrt{4 \cdot 11} = 2 \sqrt{11}
\][/tex]
Substituting this in:
[tex]\[
x = \frac{2 \pm 2 \sqrt{11} \cdot i}{6}
\][/tex]
Divide the numerator and denominator by 2:
[tex]\[
x = \frac{1 \pm \sqrt{11} \cdot i}{3}
\][/tex]
Thus, the solutions to the equation [tex]\(3x^2 - 2x + 4 = 0\)[/tex] are:
[tex]\[
x = \frac{1 \pm i\sqrt{11}}{3}
\][/tex]
Therefore, the correct option is:
B. [tex]\(x = \frac{1 \pm i\sqrt{11}}{3}\)[/tex]
