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.

Question

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 \(x\):

1. Setting the first factor equal to zero:
\[
46x = 0
\]
Dividing both sides by 46:
\[
x = 0
\]

2. 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:
\[
x = 0 \quad \text{and} \quad x = 3
\]

Answer 1

Your steps and calculations are mostly correct, but there’s a small error in your factorization of the quadratic equation. Let's go through it again for clarity.

Given the equation: \[ 46x^2 - 138x = 0 \]

1. Identify the Greatest Common Factor (GCF):

   The GCF of \(46x^2\) and \(-138x\) is indeed \(46x\).

2. Factor out the GCF: When you factor \(46x\) out of the equation, it should be done correctly: \[ 46x(x - 3) = 0 \]

3. Set each factor equal to zero:

   • For the first factor: \[ 46x = 0 \] Dividing both sides by 46 gives: \[ x = 0 \]

   • For the second factor: \[ x - 3 = 0 \] Adding 3 to both sides yields: \[ x = 3 \]

4. Conclusion:

   Thus, the solutions to the quadratic equation \(46x^2 - 138x = 0\) are: \[ x = 0 \quad \text{and} \quad x = 3 \]

So your final results are correct: \[ x = 0 \quad \text{and} \quad x = 3 \]

Well done!