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(x-1)^x+2=(x-1)^x+6

2 min read Jun 17, 2024
(x-1)^x+2=(x-1)^x+6

Solving the Equation: (x-1)^x+2 = (x-1)^x+6

This equation may appear complex at first glance, but it can be solved with a simple understanding of exponents and algebraic manipulation.

Simplifying the Equation

  1. Recognize the Common Factor: Both terms on each side of the equation have a common factor of (x-1)^x.

  2. Factor out the Common Factor: (x-1)^x + 2 = (x-1)^x + 6 can be rewritten as: (x-1)^x * (1 + 2/(x-1)^x) = (x-1)^x * (1 + 6/(x-1)^x)

  3. Cancel the Common Factor: Since both sides have (x-1)^x as a factor, we can cancel it out, leaving us with: 1 + 2/(x-1)^x = 1 + 6/(x-1)^x

  4. Isolate the Variable Term: Subtracting 1 from both sides, we get: 2/(x-1)^x = 6/(x-1)^x

  5. Solve for (x-1)^x: Multiplying both sides by (x-1)^x and simplifying, we get: 2 = 6 This equation is not true for any value of x.

Conclusion

Therefore, there is no solution to the equation (x-1)^x+2 = (x-1)^x+6. This indicates that the initial assumption of having a common factor led to an impossible result, meaning the original equation is not solvable.

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