Certainly! Let's solve the equation [tex]\((x+5)(x-3)=0\)[/tex] step-by-step.
### Step 1: Understand the Problem
The equation [tex]\((x+5)(x-3)=0\)[/tex] is a product of two binomials. To find the solutions, we need to figure out the values of [tex]\(x\)[/tex] that make this equation true.
### Step 2: Apply 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. Therefore, for [tex]\((x+5)(x-3)=0\)[/tex], either:
[tex]\[ x + 5 = 0 \][/tex]
or
[tex]\[ x - 3 = 0 \][/tex]
### Step 3: Solve Each Equation
Let's solve each of these equations individually:
1. First Equation:
[tex]\[ x + 5 = 0 \][/tex]
To find [tex]\(x\)[/tex], subtract 5 from both sides:
[tex]\[ x = -5 \][/tex]
2. Second Equation:
[tex]\[ x - 3 = 0 \][/tex]
To find [tex]\(x\)[/tex], add 3 to both sides:
[tex]\[ x = 3 \][/tex]
### Step 4: Conclude the Solutions
The solutions to the equation [tex]\((x+5)(x-3)=0\)[/tex] are [tex]\( x = -5 \)[/tex] and [tex]\( x = 3 \)[/tex].
### Verification
Just to verify, if we plug these values back into the original equation, both should satisfy it:
1. For [tex]\( x = -5 \)[/tex]:
[tex]\[ (-5 + 5)(-5 - 3) = 0 \cdot (-8) = 0 \][/tex]
2. For [tex]\( x = 3 \)[/tex]:
[tex]\[ (3 + 5)(3 - 3) = 8 \cdot 0 = 0 \][/tex]
Both values satisfy the equation, verifying our solutions.
### Conclusion
The solutions to the equation [tex]\((x+5)(x-3)=0\)[/tex] are:
[tex]\[ x = -5 \quad \text{and} \quad x = 3 \][/tex]
