Sure, let’s solve the equation step by step and find the product of its solutions.
The given equation is:
[tex]\[
(2y - 1)(8 - y) = 0
\][/tex]
For the product of two factors to be zero, at least one of the factors must be zero. Hence, we set each factor equal to zero and solve for [tex]\(y\)[/tex]:
1. Set the first factor to zero:
[tex]\[
2y - 1 = 0
\][/tex]
Add 1 to both sides:
[tex]\[
2y = 1
\][/tex]
Divide by 2:
[tex]\[
y = \frac{1}{2}
\][/tex]
So, [tex]\(u = \frac{1}{2}\)[/tex].
2. Set the second factor to zero:
[tex]\[
8 - y = 0
\][/tex]
Add [tex]\(y\)[/tex] to both sides:
[tex]\[
8 = y
\][/tex]
So, [tex]\(d = 8\)[/tex].
Now we have the two solutions: [tex]\( u = \frac{1}{2} \)[/tex] and [tex]\( d = 8 \)[/tex].
To find the value of [tex]\( u \cdot d \)[/tex], we multiply these solutions:
[tex]\[
u \cdot d = \left(\frac{1}{2}\right) \cdot 8
\][/tex]
Simplify the multiplication:
[tex]\[
u \cdot d = \frac{1}{2} \cdot 8 = 4
\][/tex]
Thus, the value of [tex]\( u \cdot d \)[/tex] is:
[tex]\[
\boxed{4}
\][/tex]
