1. The problem asks if the quadratic equation $3x^2 - 2x + 1 = 0$ has real roots.
2. To determine this, we calculate the discriminant $\Delta $ using the formula:
$$ \Delta = b^2 - 4ac $$
where $a=3$, $b=-2$, and $c=1$.
3. Substitute the values:
$$ \Delta = (-2)^2 - 4 \times 3 \times 1 = 4 - 12 = -8 $$
4. Since the discriminant $\Delta = -8$ is less than zero, the quadratic equation has no real roots.
5. Therefore, the answer is **NO**, the equation does not have real roots because the parabola does not intersect the x-axis.
Final answer: NO
Real Roots
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