Question
$$x-\frac{x+7}{x+8} = \frac{x}{x+8}$$
Answer
$$ \begin{matrix}x_1 = 1 & x_2 = -7 \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} x-\frac{x+7}{x+8} &= \frac{x}{x+8}&& \text{multiply ALL terms by } \color{blue}{ x+8 }. \\[1 em](x+8)x-(x+8)\frac{x+7}{x+8} &= (x+8)\frac{x}{x+8}&& \text{cancel out the denominators} \\[1 em]x^2+8x-(x+7) &= x&& \text{simplify left side} \\[1 em]x^2+8x-x-7 &= x&& \\[1 em]x^2+7x-7 &= x&& \text{move all terms to the left hand side } \\[1 em]x^2+7x-7-x &= 0&& \text{simplify left side} \\[1 em]x^2+6x-7 &= 0&& \\[1 em] \end{aligned} $$
$ x^{2}+6x-7 = 0 $ is a quadratic equation.
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