解の公式を導く

\begin{eqnarray} ax^{2} + bx + c &=& 0\\ a \left( x^{2} + \frac{b}{a}x \right) + c &=& 0\\ a\left\{ \left(x + \frac{b}{2a} \right)^{2} - \frac{b^{2}}{4a^{2}} \right\} + c &=& 0\\ a\left( x + \frac{b}{2a} \right)^{2} - \frac{b^{2}}{4a} + c &=& 0\\ a\left( x + \frac{b}{2a} \right)^{2} - \frac{-b^{2}+4ac}{4a} + c &=& 0\\ a\left( x + \frac{b}{2a} \right)^{2} &=& \frac{-b^{2}+4ac}{4a} + c\\ \left( x + \frac{b}{2a} \right)^{2} &=& \frac{-b^{2}+4ac}{4a^{2}} + c\\ x + \frac{b}{2a} &=& \pm \sqrt{\frac{b^{2} - 4ac}{4a^{2}}}\\ x + \frac{b}{2a} &=& \pm \frac{\sqrt{b^{2}-4ac}}{2a}\\ x &=& - \frac{b}{2a} \pm \frac{\sqrt{b^{2}-4ac}}{2a}\\ x &=& \frac{-b\pm \sqrt{b^{2}-4ac}}{2a} \end{eqnarray}