Derivation of the abc Quadratic Formula

The general quadratic equation  
\[ax^2 +bx+c=0\]
  can be solved for  
  using the formula  
\[x= \frac{-b \pm \sqrt{b^2-4ac}}{2a}\]
We can derive this formula by completing the square.
Starting from  
\[ax^2 +bx+c=0\]
, multiply by  
\[a^2x^2 +abx+ac=0\]

  for the moment and complete the square for  
\[a^2x^2 +abx\]
\[a^2x^2 +abx+ac=((ax+\frac{b}{2})^2 - (\frac{b}{2})^2)+ac=0 \]

Now add  
  and subtract  
\[(ax+\frac{b}{2})^2 = (\frac{b}{2})^2-ac =\frac{b^2}{4}-ac= \frac{b^2-4ac}{4}\]

Square root both sides.
\[ax+\frac{b}{2} = \sqrt{(\frac{b^2-4ac}{4})}=\pm \frac{\sqrt{b^2-4ac}}{2}\]

Now subtract  
  from both sides.
\[ax =- \frac{b}{2} \sqrt{(\frac{b^2-4ac}{4})}= \frac{-b \pm\sqrt{b^2-4ac}}{2}=\]

Finally divide both sides by  
\[x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\]

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