(2x-1)2=(x+1)2 Quadratic Equation

2 min read Jun 16, 2024

Solving the Quadratic Equation: (2x-1)² = (x+1)²

This article will walk you through the steps of solving the quadratic equation (2x-1)² = (x+1)².

Understanding the Problem

We're given an equation with squared terms on both sides. Our goal is to find the values of x that satisfy the equation.

Solving the Equation

Expand the squares:
- (2x-1)² = 4x² - 4x + 1
- (x+1)² = x² + 2x + 1
Set up the equation:
- 4x² - 4x + 1 = x² + 2x + 1
Simplify by combining like terms:
- 3x² - 6x = 0
Factor out a common factor:
- 3x(x - 2) = 0
Set each factor equal to zero:
- 3x = 0 or x - 2 = 0
Solve for x:
- x = 0 or x = 2

The Solutions

Therefore, the solutions to the quadratic equation (2x-1)² = (x+1)² are x = 0 and x = 2.

Verification

To verify our solutions, we can substitute each value of x back into the original equation:

For x = 0: (2(0)-1)² = (-1)² = 1; (0+1)² = 1² = 1. The equation holds true.
For x = 2: (2(2)-1)² = (3)² = 9; (2+1)² = 3² = 9. The equation holds true.

Conclusion

We have successfully solved the quadratic equation (2x-1)² = (x+1)² by expanding, simplifying, factoring, and solving for x. The solutions are x = 0 and x = 2, which can be verified by plugging them back into the original equation.