Certainly! Let's solve the equation step-by-step.
We are given the equation:
[tex]\[ w(w-2)(6w + 5) = 0 \][/tex]
To find the solutions, we need to determine the values of [tex]\( w \)[/tex] that make the equation equal to zero. This is a factored polynomial equation, and we can use the Zero Product Property, which states that if a product of factors is zero, then at least one of the factors must be zero.
Let's consider each factor individually:
1. First factor: [tex]\( w \)[/tex]
We set the first factor equal to zero:
[tex]\[ w = 0 \][/tex]
Therefore, one solution is:
[tex]\[ w = 0 \][/tex]
2. Second factor: [tex]\( w - 2 \)[/tex]
We set the second factor equal to zero:
[tex]\[ w - 2 = 0 \][/tex]
Solving for [tex]\( w \)[/tex], we get:
[tex]\[ w = 2 \][/tex]
Therefore, another solution is:
[tex]\[ w = 2 \][/tex]
3. Third factor: [tex]\( 6w + 5 \)[/tex]
We set the third factor equal to zero:
[tex]\[ 6w + 5 = 0 \][/tex]
Solving for [tex]\( w \)[/tex], we subtract 5 from both sides:
[tex]\[ 6w = -5 \][/tex]
Then, we divide both sides by 6:
[tex]\[ w = -\frac{5}{6} \][/tex]
Therefore, the third solution is:
[tex]\[ w = -\frac{5}{6} \][/tex]
Putting it all together, the solutions to the equation [tex]\( w(w-2)(6w+5) = 0 \)[/tex] are:
[tex]\[ w = 0, \quad w = 2, \quad \text{and} \quad w = -\frac{5}{6} \][/tex]
Hence, the complete set of solutions is:
[tex]\[ \boxed{[-\frac{5}{6}, 0, 2]} \][/tex]
