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