The given equation is a quadratic equation in the form $ax^2 + bx + c = 0$, where:

• $a = 3$
• $b = -5$
• $c = 1$

To solve for $x$, we can use the quadratic formula:

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

Now, plug in the values of $a$, $b$, and $c$:

$x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(3)(1)}}{2(3)}$

Simplifying step by step:

$x = \frac{5 \pm \sqrt{25 - 12}}{6}$

$x = \frac{5 \pm \sqrt{13}}{6}$

So, the two possible solutions for $x$ are:

$x = \frac{5 + \sqrt{13}}{6} \quad \text{or} \quad x = \frac{5 - \sqrt{13}}{6}$

These are the exact solutions. If you'd like the decimal approximation:

$x_1 \approx 1.434 \quad \text{and} \quad x_2 \approx 0.233$

Would you like more details or clarifications?

Here are 5 related questions:

1. How do you determine whether a quadratic equation has real or complex solutions?
2. What does the discriminant in the quadratic formula tell you about the solutions?
3. Can you graph the quadratic equation $3x^2 - 5x + 1 = 0$?
4. How would you solve a quadratic equation using factoring or completing the square?
5. What are the properties of quadratic functions and their graphs?

Tip: The discriminant $b^2 - 4ac$ helps you know if solutions are real or complex: if positive, the solutions are real; if zero, there's exactly one real solution; if negative, the solutions are complex.