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Solving the Equation: (x+1)^-1 = x^-1 + x
This article explores the equation (x+1)^-1 = x^-1 + x and analyzes its solutions.
Understanding the Equation
Firstly, let's break down the equation:
- (x+1)^-1: This represents the reciprocal of (x+1), which is equivalent to 1/(x+1).
- x^-1: This represents the reciprocal of x, which is equivalent to 1/x.
Therefore, the equation can be rewritten as:
(1/(x+1)) = (1/x) + x
Solving for x
To solve for x, we can follow these steps:
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Get rid of the fractions: Multiply both sides of the equation by x(x+1) to clear the denominators:
x = (x+1) + x^2(x+1)
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Simplify and rearrange: Expand the right side, then move all terms to one side:
x = x + 1 + x^3 + x^2 0 = x^3 + x^2 - 1
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Solve the cubic equation: Finding the exact solutions for this cubic equation is not straightforward. We can use numerical methods or graphing calculators to find approximate solutions.
Analyzing the Solutions
The cubic equation has one real solution and two complex solutions. The real solution lies between -2 and -1.
Conclusion
The equation (x+1)^-1 = x^-1 + x has a single real solution and two complex solutions. While finding the exact value of the real solution requires numerical methods, we can understand its approximate location and the nature of the solutions.