To derive the equations from the given quadratic equation using the zero product property, follow these steps:
1. Recognize that the original equation is:
[tex]\[
6(x + 7)(x - 4) = 0
\][/tex]
2. The zero product property states that if the product of several factors equals zero, then at least one of the factors must be zero.
3. Identify the factors in the equation:
[tex]\[
(x + 7) \quad \text{and} \quad (x - 4)
\][/tex]
4. Set each factor equal to zero to find the two equations.
5. For the first factor [tex]\( (x + 7) \)[/tex]:
[tex]\[
x + 7 = 0
\][/tex]
6. For the second factor [tex]\( (x - 4) \)[/tex]:
[tex]\[
x - 4 = 0
\][/tex]
The equations derived from the factored form of the quadratic equation are:
[tex]\[
\boxed{x + 7 = 0}
\][/tex]
[tex]\[
\boxed{x - 4 = 0}
\][/tex]
