To solve the equation [tex]\(2(v-3)(-v+1) = 0\)[/tex], follow these steps:
1. Identify the factored form: The given equation is already in a factored form. The zero-product property states that if a product of factors equals zero, at least one of the factors must be zero.
2. Set each factor to zero and solve for [tex]\(v\)[/tex]:
- The first factor is [tex]\(v-3\)[/tex]:
[tex]\[
v - 3 = 0
\][/tex]
Solving for [tex]\(v\)[/tex] gives:
[tex]\[
v = 3
\][/tex]
- The second factor is [tex]\(-v+1\)[/tex]:
[tex]\[
-v + 1 = 0
\][/tex]
Solving for [tex]\(v\)[/tex] gives:
[tex]\[
-v = -1 \implies v = 1
\][/tex]
3. List all solutions: Combining the solutions from both factors, the solutions to the equation are:
[tex]\[
v = 3, 1
\][/tex]
Therefore, the values of [tex]\(v\)[/tex] that satisfy the equation [tex]\(2(v-3)(-v+1) = 0\)[/tex] are [tex]\(v = 3\)[/tex] and [tex]\(v = 1\)[/tex].
