$$\frac{1}{x}+\frac{1}{x+1}+\frac{1}{x+2} = 2$$
Answer
$$ \begin{matrix}x_1 = 0.88923 & x_2 = -0.64458 & x_3 = -1.74464 \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{1}{x}+\frac{1}{x+1}+\frac{1}{x+2} &= 2&& \text{multiply ALL terms by } \color{blue}{ x(x+1)(x+2) }. \\[1 em]x(x+1)(x+2)\cdot\frac{1}{x}+x(x+1)(x+2)\cdot\frac{1}{x+1}+x(x+1)(x+2)\cdot\frac{1}{x+2} &= x(x+1)(x+2)\cdot2&& \text{cancel out the denominators} \\[1 em]x^2+3x+2+x^2+2x+x^2+x &= 2x^3+6x^2+4x&& \text{simplify left side} \\[1 em]2x^2+5x+2+x^2+x &= 2x^3+6x^2+4x&& \\[1 em]3x^2+6x+2 &= 2x^3+6x^2+4x&& \text{move all terms to the left hand side } \\[1 em]3x^2+6x+2-2x^3-6x^2-4x &= 0&& \text{simplify left side} \\[1 em]-2x^3-3x^2+2x+2 &= 0&& \\[1 em] \end{aligned} $$
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using qubic formulas.
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