To solve the equation \(0 = 3x^2 - 4x + 1\) by factoring, we can start by rewriting the equation into a factored form.
First, let's look for two numbers that multiply to \(ac\) (where \(a = 3\) and \(c = 1\), thus \(ac = 3 \cdot 1 = 3\)) and add up to \(b\) (where \(b = -4\)).
We need two numbers that multiply to \(3\) and add to \(-4\). The numbers that work are \(-3\) and \(-1\).
We can rewrite the middle term of the quadratic equation using these two numbers:
\[ 3x^2 - 3x - 1x + 1 = 0 \]
Next, we can group the terms:
\[ (3x^2 - 3x) + (-1x + 1) = 0 \]
Now factor out the common factors in each group:
\[ 3x(x - 1) - 1(x - 1) = 0 \]
This allows us to factor by grouping:
\[ (3x - 1)(x - 1) = 0 \]
Now we can set each factor equal to zero to find the solutions:
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\(3x - 1 = 0\) \[ 3x = 1 \implies x = \frac{1}{3} \]
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\(x - 1 = 0\) \[ x = 1 \]
Thus, the solutions to the equation \(0 = 3x^2 - 4x + 1\) are \(x = \frac{1}{3}\) and \(x = 1\).
Since one solution was given, we can conclude that the other solution is:
\[ \boxed{1} \]