To solve the equation \( 46x^2 - 138x = 0 \) by factoring out the greatest common factor (GCF), we first identify the GCF of the terms \( 46x^2 \) and \( -138x \).
- The GCF of \( 46 \) and \( 138 \) is \( 46 \).
- Both terms also have a factor of \( x \).
Thus, the GCF of the entire expression is \( 46x \).
Now, we can factor out the GCF from the equation:
\[ 46x(x - 3) = 0 \]
Next, we will set each factor equal to zero to solve for \( x \):
-
\( 46x = 0 \)
- Dividing both sides by \( 46 \) gives: \[ x = 0 \]
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\( x - 3 = 0 \)
- Adding \( 3 \) to both sides gives: \[ x = 3 \]
Thus, the solutions to the equation \( 46x^2 - 138x = 0 \) are:
\[ \boxed{0} \quad \text{and} \quad \boxed{3} \]
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