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Answer:
Understanding the Problem
Problem:
* Raise x to the negative one power.
* Set the result equal to x - 2.
Equation:
* x^(-1) = x - 2
Solving the Equation
To solve this equation, we can manipulate it to get a polynomial equation and then find its roots.
Step 1: Convert to a fraction
* x^(-1) is equivalent to 1/x.
* So, the equation becomes: 1/x = x - 2
Step 2: Clear the fraction
* Multiply both sides by x:
* 1 = x^2 - 2x
Step 3: Rearrange into a quadratic equation
* Subtract 1 from both sides:
* x^2 - 2x - 1 = 0
Step 4: Solve the quadratic equation
* This equation cannot be easily factored, so we'll use the quadratic formula:
* x = [-b ± sqrt(b^2 - 4ac)] / (2a)
* Where a = 1, b = -2, and c = -1
* Plugging in the values, we get:
* x = [2 ± sqrt((-2)^2 - 4(1)(-1))] / (2*1)
* x = [2 ± sqrt(8)] / 2
* x = 1 ± sqrt(2)
Solutions:
* x = 1 + sqrt(2)
* x = 1 - sqrt(2)
Therefore, the solutions to the equation x^(-1) = x - 2 are x = 1 + sqrt(2) and x = 1 - sqrt(2).