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Solving the Equation: (x+1/x)^3=125
In this article, we will explore the solution to the equation (x+1/x)^3=125. This equation involves the use of algebraic expressions and properties of exponential functions.
Understanding the Equation
The given equation is (x+1/x)^3=125. Here, we have a binomial expression (x+1/x) raised to the power of 3, and it is equal to 125. Our goal is to find the value(s) of x that satisfy this equation.
Simplifying the Equation
To simplify the equation, we can start by cubing the binomial expression x+1/x. Using the formula for the cube of a binomial, we get:
(x+1/x)^3 = x^3 + 3x^2(1/x) + 3x(1/x)^2 + (1/x)^3
Simplifying the expression, we get:
(x^3 + 1/x^3) + 3(x + 1/x) = 125
Solving for x
Now, we can see that the equation is a quadratic equation in terms of x + 1/x. Let's substitute y = x + 1/x. Then, the equation becomes:
y^3 - 3y - 125 = 0
This is a cubic equation in y. We can solve for y using numerical methods or algebraic methods. Using numerical methods, we get:
y = 5
Now, we need to solve for x. Substituting y = x + 1/x, we get:
x + 1/x = 5
Multiplying both sides by x, we get:
x^2 - 5x + 1 = 0
Solving this quadratic equation, we get:
x = (5 ± √21)/2
Simplifying, we get:
x = (5 + √21)/2 or x = (5 - √21)/2
Conclusion
In conclusion, we have solved the equation (x+1/x)^3=125 and found the values of x that satisfy the equation. The solutions are x = (5 + √21)/2 and x = (5 - √21)/2.
