Solving the quadratic

Quadratic equation:

(1)
\begin{equation} ax^2+bx+c=0 \end{equation}

To solve the quadratic:

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