To solve the equation [tex]\( w(w-2)(6w+5) = 0 \)[/tex], we need to identify the values of [tex]\( w \)[/tex] that make this equation true. The product of several factors equals zero if and only if at least one of the factors is zero. Therefore, we will set each factor equal to zero and solve for [tex]\( w \)[/tex].
1. The first factor is [tex]\( w \)[/tex]:
[tex]\[
w = 0
\][/tex]
This gives us the first solution [tex]\( w = 0 \)[/tex].
2. The second factor is [tex]\( w - 2 \)[/tex]:
[tex]\[
w - 2 = 0
\][/tex]
Solving for [tex]\( w \)[/tex], we get:
[tex]\[
w = 2
\][/tex]
This gives us the second solution [tex]\( w = 2 \)[/tex].
3. The third factor is [tex]\( 6w + 5 \)[/tex]:
[tex]\[
6w + 5 = 0
\][/tex]
Solving for [tex]\( w \)[/tex], we get:
[tex]\[
6w = -5 \implies w = -\frac{5}{6}
\][/tex]
This gives us the third solution [tex]\( w = -\frac{5}{6} \)[/tex].
Thus, the complete set of solutions to the equation [tex]\( w(w-2)(6w+5) = 0 \)[/tex] is:
[tex]\[
w = 0, \quad w = 2, \quad w = -\frac{5}{6}
\][/tex]
These are the values of [tex]\( w \)[/tex] that satisfy the given equation.
