The given equation is:

$3x^2 - x + 4 = 0$

This is a quadratic equation in the form $ax^2 + bx + c = 0$. Here, $a = 3$, $b = -1$, and $c = 4$.

Step-by-Step Solution:

To solve this quadratic equation, we can use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Substitute the values of $a$, $b$, and $c$:

$x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(3)(4)}}{2(3)}$

$x = \frac{1 \pm \sqrt{1 - 48}}{6}$

$x = \frac{1 \pm \sqrt{-47}}{6}$

Since the discriminant is negative ($\sqrt{-47}$), the solutions will be complex. Let's simplify:

$x = \frac{1 \pm i\sqrt{47}}{6}$

Thus, the two solutions are:

$x = \frac{1 + i\sqrt{47}}{6} \quad \text{and} \quad x = \frac{1 - i\sqrt{47}}{6}$

These are the complex solutions for the quadratic equation.

Would you like more details or have any further questions?

Here are some related questions you might find helpful:

1. How can I interpret complex solutions geometrically?
2. Can a quadratic equation always have real solutions?
3. What is the discriminant and how does it determine the nature of the solutions?
4. How do you solve quadratic equations by factoring?
5. What is the relationship between the coefficients of a quadratic equation and its solutions?

Tip: The discriminant, $b^2 - 4ac$, is key to determining whether the roots of a quadratic equation are real or complex.