Sure, let's solve the equation [tex]\((x-3)(x+11)=0\)[/tex] step by step.
1. Understand the Zero-Product Property:
The Zero-Product Property states that if the product of two factors is zero, at least one of the factors must be zero. This means if [tex]\(a \cdot b = 0\)[/tex], then either [tex]\(a = 0\)[/tex] or [tex]\(b = 0\)[/tex].
2. Apply the Zero-Product Property:
For the equation [tex]\((x-3)(x+11) = 0\)[/tex], we set each factor equal to zero and solve for [tex]\(x\)[/tex]:
- First factor: [tex]\(x - 3 = 0\)[/tex]
- Second factor: [tex]\(x + 11 = 0\)[/tex]
3. Solve the First Equation: [tex]\(x - 3 = 0\)[/tex]
To solve for [tex]\(x\)[/tex], add 3 to both sides of the equation:
[tex]\[
x - 3 + 3 = 0 + 3
\][/tex]
[tex]\[
x = 3
\][/tex]
4. Solve the Second Equation: [tex]\(x + 11 = 0\)[/tex]
To solve for [tex]\(x\)[/tex], subtract 11 from both sides of the equation:
[tex]\[
x + 11 - 11 = 0 - 11
\][/tex]
[tex]\[
x = -11
\][/tex]
5. Combine the Solutions:
Therefore, the solutions to the equation [tex]\((x-3)(x+11)=0\)[/tex] are [tex]\(x = 3\)[/tex] and [tex]\(x = -11\)[/tex].
So, the solutions are:
[tex]\[
\boxed{3 \text{ and } -11}
\][/tex]
