\[ ax^2 + bx + c = 0 \]
\[ x^2 + \frac{bx}{a} + \frac{c}{a} = 0 \]
\[ x^2 + \frac{bx}{a}=-\frac{c}{a} \]
\[ x^2 + \frac{bx}{a} + (\frac{b}{2a})^2=-\frac{c}{a}+(\frac{b}{2a})^2 \]
\[ (x + \frac{b}{2a})^2 = -\frac{c}{a}+(\frac{b}{2a})^2 \]
\[ x + \frac{b}{2a} = \pm \sqrt{-\frac{c}{a}+(\frac{b}{2a})^2} = \pm\sqrt{-\frac{-4ac}{4a^2} + \frac{b^2}{4a^2}}=\pm\sqrt{\frac{b^2-4ac}{4a^2}}=\pm\frac{\sqrt{b^2-4ac}}{2a} \]
\[ x = -\frac{b}{2a}\pm\frac{\sqrt{b^2-4ac}}{2a} \]
\[ x = \frac{-b\pm\sqrt{b^2-4ac}}{2a} \]