Solving an equation over the reals: $ x^3 + 1 = 2\sqrt[3]{{2x - 1}}$

We have $$x^3+1=2(2x-1)^{1/3}\iff x^3=2(2x-1)^{1/3} -1.$$ Here, setting $y=(2x-1)^{1/3}$ gives us $$y^3=2x-1 \ \ \text{and}\ \ x^3=2y-1.$$ Hence, we have $$\begin{align}y^3-x^3=(2x-1)-(2y-1)&\Rightarrow (y-x)(y^2+yx+x^2)=2(x-y)\\&\Rightarrow (y-x)(y^2+yx+x^2+2)=0\\&\Rightarrow (y-x)\{(x+(y/2))^2 + (3/4)y^2+2\}=0\\&\Rightarrow y=x.\end{align}$$ Hence, we have $$x^3=2x-1\iff (x-1)(x^2+x-1)=0\iff x=1,\frac{-1\pm\sqrt 5}{2}.$$

This solution will be frustratingly incomplete, since I'm not seeing an easy way to solve completely by hand. But here is where the road leads...

When solving equations in $x$, it is usually helpful to rewrite it in terms of a polynomial in $x$. To that end, cube both sides to obtain $8(2x-1)=(x^3+1)^3$, which upon moving terms to one side and expanding yields $$x^9+3x^6+3x^3-16x+9=0$$ Noting that $x=1$ is a trivial solution, we see that we can factorize this as $$(x-1)(x^8+x^7+x^6+4x^5+4x^4+4x^3+7x^2+7x-1)=0$$ One can in fact factor this further, though I should confess I only saw this after looking up the roots: $$(x-1)(x^2+x-1)(x^6+2x^4+2x^3+4x^2+2x+9)=0$$ The roots of the first two factors give three real roots which can be found by hand as $x=1,(-1\pm\sqrt{5})/2$. It turns out that the last factor has no real roots, and so these three are the result.

The two incomplete points:

1. Does anyone know a good way to spot the factor of $x^2+x-1$?
2. Does anyone see an obvious way to verify the lack of real roots in the last equation?

$\ \frac{{x^3 + 1}}{2} = \sqrt[3]{{2x - 1}} \ $, denote LHS by $f(x)$ .

Since $f(x)$ is bijective, it must have an inverse function, particularly the RHS in this situation.So, the equation is simplified to $\ f(x) = f^{ - 1} (x) \ $, which further yields $\ f(f(x)) = x \ $. Since both functions are strictly increasing, it would only make sense that $f(x)=x$, therefore $\ x^3 + 1 - 2x = 0 \ $. $x=1$ is a trivial solution so we may rewrite our equation as $\ (x - 1)(x^2 + x - 1) = 0 \ $ .

By solving the quadratic we get $\ x_{1,2} = \frac{{ - 1 \pm \sqrt 5 }}{2} \ $ .We conclude that the real solutions of our equation are comprised by $\ S = \{ 1,\frac{{ - 1 \pm \sqrt 5 }}{2}\} \ $ .

For $x$ - real $$2\cdot(2x-1)^\frac13=x^3+1$$

Let $y= (2x-1)^\frac13$

Therefore, $$y^3=2x-1$$ $$x=\frac{(y^3+1)}{2}$$

Then you get $$2\cdot y=\left(\frac{y^3+1}2\right)^3+1$$

Resolve it for $y$, and then replace find the $x$

Resolving for $y$:

1. multiply both sides by 8 and you get

$$16\cdot y=(y^3+1)^3+8$$

$$16\cdot y = y^9+ 3\cdot y^6+3\cdot y^3 +9$$

$$(y-1)(y^6 + 2\cdot y^4+ 2\cdot y^3 +4\cdot y^2+2\cdot y+9 )(y^2 + y - 1)=0$$

Then we get: $y-1 = 0 \implies y=1$

$y^2 +y - 1=0 \implies y = \frac{1}{2}(-1\pm \sqrt5)$

$y^6 + 2\cdot y^4+ 2\cdot y^3 +4\cdot y^2+2\cdot y+9 =0\implies$ No roots