Sure, let's solve the equation [tex]\(3w^2 + 27w = 0\)[/tex] step by step.
1. Factor out the common term:
The first step is to factor out the greatest common factor from the terms. Notice that both terms have a common factor of [tex]\(3w\)[/tex].
[tex]\[
3w^2 + 27w = 3w(w + 9)
\][/tex]
2. Set each factor to zero:
According to the Zero Product Property, if the product of two factors is zero, then at least one of the factors must be zero. Therefore, we can set each factor equal to zero and solve for [tex]\(w\)[/tex].
[tex]\[
3w = 0 \quad \text{or} \quad w + 9 = 0
\][/tex]
3. Solve each equation:
- For [tex]\(3w = 0\)[/tex], divide both sides by 3:
[tex]\[
w = 0
\][/tex]
- For [tex]\(w + 9 = 0\)[/tex], subtract 9 from both sides:
[tex]\[
w = -9
\][/tex]
4. Combine the solutions:
The solutions to the equation are [tex]\(w = 0\)[/tex] and [tex]\(w = -9\)[/tex].
So, the solutions for [tex]\(w\)[/tex] are:
[tex]\[
w = -9, 0
\][/tex]
