To solve the equation [tex]\(x(2x - 1) = 0\)[/tex], we will use the property that if a product of two factors is zero, then at least one of the factors must be zero. Let's solve it step-by-step.
### Step 1: Setting up the equation
The given equation is:
[tex]\[ x(2x - 1) = 0 \][/tex]
### Step 2: Solving for each factor
The product of two factors [tex]\(x\)[/tex] and [tex]\((2x - 1)\)[/tex] is equal to zero. Thus, we set each factor equal to zero and solve for [tex]\(x\)[/tex]:
1. First factor:
[tex]\[ x = 0 \][/tex]
This gives us one solution:
[tex]\[ x = 0 \][/tex]
2. Second factor:
[tex]\[ 2x - 1 = 0 \][/tex]
To solve for [tex]\(x\)[/tex], we need to isolate [tex]\(x\)[/tex]:
[tex]\[ 2x = 1 \][/tex]
[tex]\[ x = \frac{1}{2} \][/tex]
This gives us another solution:
[tex]\[ x = \frac{1}{2} \][/tex]
### Step 3: Combine the solutions
The solutions to the equation are:
[tex]\[ x = 0 \][/tex]
and
[tex]\[ x = \frac{1}{2} \][/tex]
### Conclusion
The equation [tex]\( x(2x - 1) = 0 \)[/tex] has two solutions:
[tex]\[ x = 0 \quad \text{and} \quad x = \frac{1}{2} \][/tex]
These steps outline the process to find that the solutions to the equation [tex]\( x(2x - 1) = 0 \)[/tex] are [tex]\( x = 0 \)[/tex] and [tex]\( x = \frac{1}{2} \)[/tex].
