Sure, let's solve the equation [tex]\((x + 3)(x - 5) = 0\)[/tex] step by step.
### Step 1: Understand the Zero Product Property
The Zero Product Property states that if the product of two factors is zero, then at least one of the factors must be zero. This can be written as:
[tex]\[
a \cdot b = 0 \implies a = 0 \text{ or } b = 0
\][/tex]
### Step 2: Apply the Zero Product Property
Given [tex]\((x + 3)(x - 5) = 0\)[/tex], we have two factors, [tex]\(x + 3\)[/tex] and [tex]\(x - 5\)[/tex]. According to the Zero Product Property, either one (or both) of these factors must be zero. Therefore, we can set each factor equal to zero and solve for [tex]\(x\)[/tex].
### Step 3: Solve Each Equation
1. Set the first factor equal to zero:
[tex]\[
x + 3 = 0
\][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[
x = -3
\][/tex]
2. Set the second factor equal to zero:
[tex]\[
x - 5 = 0
\][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[
x = 5
\][/tex]
### Step 4: Combine the Solutions
The solutions to the equation [tex]\((x + 3)(x - 5) = 0\)[/tex] are the values of [tex]\(x\)[/tex] that we found from solving each individual factor. Thus, the solutions are:
[tex]\[
x = -3 \text{ and } x = 5
\][/tex]
### Conclusion
The values of [tex]\(x\)[/tex] that satisfy the equation [tex]\((x + 3)(x - 5) = 0\)[/tex] are [tex]\(x = -3\)[/tex] and [tex]\(x = 5\)[/tex].
