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Deriving the Quadratic Formula

Solve: $ax^{2}+bx+c=0$

$x^{2}+\frac{b}{a}x+\frac{c}{a}=0$

$x^{2}+\frac{b}{a}x+(\frac{b}{2a})^{2}=-\frac{c}{a}+(\frac{b}{2a})^{2}$

$(x+\frac{b}{2a})^{2}=-\frac{c}{a}+\frac{b^{2}}{4a^2}$

$(x+\frac{b}{2a})^{2}=-\frac{c}{a}\times\frac{4a}{4a}+\frac{b^2}{4a^2}=-\frac{4ac}{4a^2}+\frac{b^2}{4a^2}$

$\sqrt{(x+\frac{b}{2a})^2}=\sqrt{\frac{b^{2}-4ac}{4a^2}}$

$x+\frac{b}{2a}=\pm\sqrt{\frac{b^{2}-4ac}{4a^2}}$

$x+\frac{b}{2a}-\frac{b}{2a}=\pm\sqrt{\frac{b^{2}-4ac}{4a^2}}-\frac{b}{2a}$

$x=-\frac{b}{2a}\pm\frac{\sqrt{b^{2}-4ac}}{2a}$

$x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$