%5E2-(x-1)%5E2%3D-2x%2B1.jpeg "(x-2)^2-(x-1)^2=-2x+1")
Solving the Equation: (x-2)² - (x-1)² = -2x + 1
This article will explore the solution to the equation (x-2)² - (x-1)² = -2x + 1. We'll use algebraic manipulation to simplify the equation and find the value(s) of x that satisfy it.
Simplifying the Equation
Let's begin by expanding the squares on the left side of the equation:
- (x-2)² = x² - 4x + 4
- (x-1)² = x² - 2x + 1
Substituting these expansions back into the original equation, we get:
(x² - 4x + 4) - (x² - 2x + 1) = -2x + 1
Simplifying further by removing the parentheses and combining like terms:
x² - 4x + 4 - x² + 2x - 1 = -2x + 1 -2x + 3 = -2x + 1
Solving for x
Notice that the x terms cancel out on both sides of the equation, leaving us with:
3 = 1
This statement is false.
Therefore, there is no solution to the equation (x-2)² - (x-1)² = -2x + 1. This indicates that there is no value of x that can satisfy the equation.
Conclusion
We have shown that the equation (x-2)² - (x-1)² = -2x + 1 has no solution. This is because the simplification process leads to a contradiction (3 = 1), implying that the original equation is inconsistent.