Question

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 \(x\):

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!