To find the solutions to the quadratic equation [tex]\(3x^2 - 5x + 4 = 0\)[/tex], we'll use the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex]. Let's identify the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] in the equation:
[tex]\[
a = 3, \quad b = -5, \quad c = 4
\][/tex]
First, compute the discriminant [tex]\(\Delta\)[/tex]:
[tex]\[
\Delta = b^2 - 4ac = (-5)^2 - 4 \cdot 3 \cdot 4 = 25 - 48 = -23
\][/tex]
The discriminant [tex]\(\Delta\)[/tex] is negative, which indicates that the quadratic equation has two complex (non-real) roots. To find the solutions, we proceed as follows:
1. Compute the square root of the negative discriminant:
[tex]\[
\sqrt{\Delta} = \sqrt{-23} = i \sqrt{23}
\][/tex]
Here, [tex]\(i\)[/tex] is the imaginary unit.
2. Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(\sqrt{\Delta}\)[/tex] into the quadratic formula:
[tex]\[
x = \frac{-(-5) \pm \sqrt{-23}}{2 \cdot 3} = \frac{5 \pm i \sqrt{23}}{6}
\][/tex]
So, the solutions to the quadratic equation [tex]\(3x^2 - 5x + 4 = 0\)[/tex] are:
[tex]\[
x = \frac{5 \pm i \sqrt{23}}{6}
\][/tex]
Comparing this result with the given options, we see that option A matches our solutions:
A. [tex]\(x = \frac{5 \pm i \sqrt{23}}{6}\)[/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{A}
\][/tex]
