$$\frac{1}{x}-\frac{1}{x+1} = 4$$
Answer
$$ \begin{matrix}x_1 = -\dfrac{ 1 }{ 2 }-\dfrac{\sqrt{ 2 }}{ 2 } & x_2 = -\dfrac{ 1 }{ 2 }+\dfrac{\sqrt{ 2 }}{ 2 } \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{1}{x}-\frac{1}{x+1} &= 4&& \text{multiply ALL terms by } \color{blue}{ x(x+1) }. \\[1 em]x(x+1)\cdot\frac{1}{x}-x(x+1)\cdot\frac{1}{x+1} &= x(x+1)\cdot4&& \text{cancel out the denominators} \\[1 em]x+1-x &= 4x^2+4x&& \text{simplify left side} \\[1 em]x+1-x &= 4x^2+4x&& \\[1 em]1 &= 4x^2+4x&& \text{move all terms to the left hand side } \\[1 em]1-4x^2-4x &= 0&& \text{simplify left side} \\[1 em]-4x^2-4x+1 &= 0&& \\[1 em] \end{aligned} $$
$ -4x^{2}-4x+1 = 0 $ is a quadratic equation.
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