%5Ex%2B2%3D(x-1)%5Ex%2B4.jpeg "(x-1)^x+2=(x-1)^x+4")
Solving the Equation (x-1)^x+2 = (x-1)^x+4
This equation might look intimidating at first glance, but we can solve it with a few simple manipulations. Let's break it down:
Simplifying the Equation
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Subtract (x-1)^x from both sides: This leaves us with: 2 = 4
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The Contradiction: The equation 2 = 4 is clearly false. This means there are no solutions to the original equation.
Understanding the Solution
The reason there are no solutions is that the expressions on both sides of the equation are fundamentally different. The terms (x-1)^x+2 and (x-1)^x+4 represent exponential functions, and for them to be equal, the bases must be the same, or the exponents must be the same. Since the bases are the same, the exponents would need to be equal as well. But they are not! This inconsistency means there can be no value of x that satisfies the equation.
In Summary
The equation (x-1)^x+2 = (x-1)^x+4 has no solutions. This is due to a fundamental mismatch between the terms on both sides of the equation.