A Nice Solution to an Interesting Equation

Problem

Solve the euqation

$\displaystyle 2\sqrt[3]{2x-1}=x^3+1$

Solution

First, divide both sides of the equation by 2, we get

$\displaystyle \sqrt[3]{2x-1}=\frac{x^3+1}{2}$

Let $f(x)=LHS=\sqrt[3]{2x-1}$ . Notice that $f^{-1}(x)=\frac{x^3+1}{2}=RHS$ , so we can just consider the equation

$\displaystyle \frac{x^3+1}{2}=x$

and get the result

$\displaystyle x_1=1, x_{2,3}=\frac{1}{2}({-1\pm \sqrt{5}})$

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