To solve the equation \( 0 = 3x^2 - 4x + 1 \) by factoring, we can start by factoring the quadratic expression.
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We can rewrite the equation: \[ 3x^2 - 4x + 1 = 0 \]
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We need to find two numbers that multiply to \( 3 \times 1 = 3 \) (the product of \( a \) and \( c \)) and add up to \( -4 \) (the coefficient of \( x \)). The two numbers that satisfy these conditions are \( -3 \) and \( -1 \).
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We can rewrite the middle term using these numbers: \[ 3x^2 - 3x - x + 1 = 0 \]
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Next, we group the terms: \[ (3x^2 - 3x) + (-x + 1) = 0 \]
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Factor out the common factors from each group: \[ 3x(x - 1) - 1(x - 1) = 0 \]
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Now, factor out the common factor \( (x - 1) \): \[ (3x - 1)(x - 1) = 0 \]
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Now we can set each factor equal to zero: \[ 3x - 1 = 0 \quad \text{or} \quad x - 1 = 0 \]
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Solving \( 3x - 1 = 0 \): \[ 3x = 1 \implies x = \frac{1}{3} \]
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Solving \( x - 1 = 0 \): \[ x = 1 \]
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So, the two solutions are: \[ x = \frac{1}{3} \quad \text{and} \quad x = 1 \]
Since the problem states that one of the solutions is \( x = \frac{1}{3} \), the other solution is: \[ \boxed{1} \]
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