Solve the quadologic equation 84x^2+ 16x = 0 by factoring out the GCF

To solve the quadratic equation \( 84x^2 + 16x = 0 \) by factoring out the greatest common factor (GCF), we first identify the GCF of the coefficients \( 84 \) and \( 16 \).

The GCF of \( 84 \) and \( 16 \) is \( 4 \). Therefore, we can factor out \( 4x \) from the equation:

\[
84x^2 + 16x = 4x(21x + 4) = 0
\]

Now, we set each factor equal to zero.

1. For the first factor:
\[
4x = 0
\]
Dividing both sides by \( 4 \), we get:
\[
x = 0
\]

2. For the second factor:
\[
21x + 4 = 0
\]
Subtracting \( 4 \) from both sides, we get:
\[
21x = -4
\]
Dividing both sides by \( 21 \), we get:
\[
x = -\frac{4}{21}
\]

Thus, the solutions to the equation \( 84x^2 + 16x = 0 \) are:
\[
x = 0 \quad \text{and} \quad x = -\frac{4}{21}
\]