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.

Learn how to solve the quadratic equation 3x^2 - 5x + 1 = 0 using the quadratic formula. This problem covers essential algebra concepts, such as the discriminant and step-by-step solution using the quadratic formula. Suitable for students in Grades 9-11.

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