Sure, let's solve the equation [tex]\((2x - 1)(x + 6) = 0\)[/tex] step by step.
1. Understand the Zero Product Property:
The zero product property states that if the product of two expressions is zero, then at least one of the expressions must be zero. This means we can set each factor to zero and solve the resulting equations separately.
2. Set Each Factor to Zero:
- The first factor is [tex]\(2x - 1\)[/tex]. Set this equal to zero:
[tex]\[
2x - 1 = 0
\][/tex]
- The second factor is [tex]\(x + 6\)[/tex]. Set this equal to zero:
[tex]\[
x + 6 = 0
\][/tex]
3. Solve Each Equation for [tex]\(x\)[/tex]:
- For the first equation:
[tex]\[
2x - 1 = 0
\][/tex]
Add 1 to both sides of the equation:
[tex]\[
2x = 1
\][/tex]
Divide both sides by 2:
[tex]\[
x = \frac{1}{2}
\][/tex]
- For the second equation:
[tex]\[
x + 6 = 0
\][/tex]
Subtract 6 from both sides:
[tex]\[
x = -6
\][/tex]
4. Write the Solutions:
The solutions to the equation [tex]\((2x - 1)(x + 6) = 0\)[/tex] are:
[tex]\[
x = \frac{1}{2} \quad \text{and} \quad x = -6
\][/tex]
Therefore, the values of [tex]\(x\)[/tex] that satisfy the given equation are [tex]\(\boxed{0.5 \text{ and } -6}\)[/tex].
