Herleitung: Quadratische Formel

\begin{align*} ax^2 + bx + c =& 0 && \vert \cdot \frac{1}{a} \\ x^2 + \frac{b}{a}x + \frac{c}{a} =& 0 \\ p \coloneqq& \frac{b}{a} \\ q \coloneqq& \frac{c}{a} \\ x^2 + px + q =& 0 && \vert - q + \frac{p^2}{4} \\ x^2 + px + \frac{p^2}{4} =& - q + \frac{p^2}{4} \\ \left(x + \frac{p}{2}\right)^2 =& \frac{p^2}{4} - q && \vert \sqrt{(...)} \\ x + \frac{p}{2} =& \pm \sqrt{\frac{p^2}{4} - q} && \vert - \frac{p}{2} \\ x =& \frac{p}{2} \pm \sqrt{\frac{p^2}{4} - q} \\ x =& \frac{b}{2a} \pm \sqrt{\frac{b^2}{4a} - \frac{c}{a}} \\ x =& \frac{b}{2a} \pm \sqrt{\frac{b^2}{4a} - \frac{4ac}{4a^2}} \\ x =& \frac{b \pm \sqrt{b^2 - 4ac}}{2a} \end{align*}