2%3D(x%2B1)2.jpeg "(2x-1)2=(x+1)2")
Solving the Equation (2x-1)² = (x+1)²
This equation involves squares, so let's break down the steps to solve it:
1. Expanding the Squares
First, we need to expand the squares on both sides of the equation using the FOIL method:
- Left Side: (2x - 1)² = (2x - 1)(2x - 1) = 4x² - 2x - 2x + 1 = 4x² - 4x + 1
- Right Side: (x + 1)² = (x + 1)(x + 1) = x² + x + x + 1 = x² + 2x + 1
Now our equation becomes: 4x² - 4x + 1 = x² + 2x + 1
2. Rearranging the Equation
Let's move all terms to one side to set the equation equal to zero:
4x² - 4x + 1 - x² - 2x - 1 = 0
Simplifying, we get: 3x² - 6x = 0
3. Factoring the Equation
We can factor out a 3x from the left side:
3x(x - 2) = 0
4. Solving for x
For the product of two factors to be zero, at least one of the factors must be zero:
- Case 1: 3x = 0 => x = 0
- Case 2: x - 2 = 0 => x = 2
Conclusion
Therefore, the solutions to the equation (2x - 1)² = (x + 1)² are x = 0 and x = 2.