$$x \cdot \frac{x+1}{2} = x^2$$
Answer
$$ \begin{matrix}x_1 = 0 & x_2 = 1 \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} x \cdot \frac{x+1}{2} &= x^2&& \text{multiply ALL terms by } \color{blue}{ 2 }. \\[1 em]2x \cdot \frac{x+1}{2} &= 2x^2&& \text{cancel out the denominators} \\[1 em]x^2+x &= 2x^2&& \text{move all terms to the left hand side } \\[1 em]x^2+x-2x^2 &= 0&& \text{simplify left side} \\[1 em]-x^2+x &= 0&& \\[1 em] \end{aligned} $$
In order to solve $ \color{blue}{ -x^{2}+x = 0 } $, first we need to factor our $ x $.
$$ -x^{2}+x = x \left( -x+1 \right) $$$ x = 0 $ is a root of multiplicity $ 1 $.
The second root can be found by solving equation $ -x+1 = 0$.
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