Question
$$\frac{1}{x}+\frac{1}{x+1} = \frac{x}{4}$$
Answer
This equation has no solution.
Explanation
$$ \begin{aligned} \frac{1}{x}+\frac{1}{x+1} &= \frac{x}{4}&& \text{multiply ALL terms by } \color{blue}{ x(x+1)\cdot4 }. \\[1 em]x(x+1)\cdot4\cdot\frac{1}{x}+x(x+1)\cdot4\cdot\frac{1}{x+1} &= x(x+1)\cdot4 \cdot \frac{x}{4}&& \text{cancel out the denominators} \\[1 em]4x+4+4x &= x^3+x^2&& \text{simplify left side} \\[1 em]8x+4 &= x^3+x^2&& \text{move all terms to the left hand side } \\[1 em]8x+4-x^3-x^2 &= 0&& \text{simplify left side} \\[1 em]-x^3-x^2+8x+4 &= 0&& \\[1 em] \end{aligned} $$
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using qubic formulas.
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Equations Solver