Question
$$\frac{x}{x}-2+32x-2 = \frac{2}{x}-2$$
Answer
$$ \begin{matrix}x_1 = \dfrac{ 1 }{ 64 }-\dfrac{\sqrt{ 257 }}{ 64 } & x_2 = \dfrac{ 1 }{ 64 }+\dfrac{\sqrt{ 257 }}{ 64 } \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{x}{x}-2+32x-2 &= \frac{2}{x}-2&& \text{multiply ALL terms by } \color{blue}{ x }. \\[1 em]x \cdot \frac{x}{x}-x\cdot2+x\cdot32x-x\cdot2 &= x\cdot\frac{2}{x}-x\cdot2&& \text{cancel out the denominators} \\[1 em]x-2x+32x^2-2x &= 2-2x&& \text{simplify left and right hand side} \\[1 em]32x^2-3x &= -2x+2&& \text{move all terms to the left hand side } \\[1 em]32x^2-3x+2x-2 &= 0&& \text{simplify left side} \\[1 em]32x^2-x-2 &= 0&& \\[1 em] \end{aligned} $$
$ 32x^{2}-x-2 = 0 $ is a quadratic equation.
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