To solve the equation [tex]\( z(2z + 1)(z + 3) = 0 \)[/tex], we need to find the values of [tex]\( z \)[/tex] that satisfy this equation. We can achieve this by factoring and using the zero-product property, which states that if the product of multiple factors equals zero, at least one of the factors must be zero.
Here’s the step-by-step solution:
1. Identify the factors:
The given equation is already factored to some extent:
[tex]\[
z(2z + 1)(z + 3) = 0
\][/tex]
2. Apply the zero-product property:
According to the zero-product property, for the product of these factors to be zero, at least one of the factors must be zero. Thus, we set each factor equal to zero separately:
[tex]\[
z = 0
\][/tex]
[tex]\[
2z + 1 = 0
\][/tex]
[tex]\[
z + 3 = 0
\][/tex]
3. Solve each equation individually:
- For [tex]\( z = 0 \)[/tex]:
This is already solved. So, one solution is:
[tex]\[
z = 0
\][/tex]
- For [tex]\( 2z + 1 = 0 \)[/tex]:
Subtract 1 from both sides of the equation:
[tex]\[
2z = -1
\][/tex]
Divide both sides by 2:
[tex]\[
z = -\frac{1}{2}
\][/tex]
- For [tex]\( z + 3 = 0 \)[/tex]:
Subtract 3 from both sides:
[tex]\[
z = -3
\][/tex]
4. Combine the solutions:
The solutions to the equation [tex]\( z(2z + 1)(z + 3) = 0 \)[/tex] are:
[tex]\[
z = -3, \quad z = -\frac{1}{2}, \quad z = 0
\][/tex]
Thus, the complete set of solutions to the equation [tex]\( z(2z + 1)(z + 3) = 0 \)[/tex] is:
[tex]\[
z = -3, -\frac{1}{2}, 0
\][/tex]
