Complete the steps of derivation of the quadratic formula

Step 1. We are going to begin with standard form a quadratic equation: [tex]ax^2+bx+c=0[/tex]

Step 2.  Divide both sides of the equation by [tex]a[/tex]:
[tex] \frac{ax^2+bx+c}{a} = \frac{0}{a} [/tex]
[tex] \frac{ax^2}{a} + \frac{bx}{c} + \frac{c}{a} =0[/tex]
[tex] x^{2} + \frac{b}{a} x+ \frac{c}{a} =0[/tex]

Step 3. Subtract [tex] \frac{c}{a} [/tex] from both sides of the equation:
[tex]x^{2} + \frac{b}{a} x+ \frac{c}{a}- \frac{c}{a} =0 - \frac{c}{a} [/tex]
[tex]x^{2} + \frac{b}{a} x=- \frac{c}{a} [/tex]

Step 4. Complete the square of the left hand side by adding [tex]( \frac{b}{2a} )^2[/tex] to both sides:
[tex]x^{2} + \frac{b}{a} x+( \frac{b}{2a} )^2=- \frac{c}{a} +( \frac{b}{2a} )^2[/tex]
[tex](x+ \frac{b}{2a} )^2=- \frac{c}{a} +( \frac{b}{2a} )^2[/tex]

Step 5. Take square root to both sides of the equation:
[tex] \sqrt{(x+ \frac{b}{2a} )^2} =+/- \sqrt{- \frac{c}{a} +( \frac{b}{2a} )^2} [/tex]
[tex]x+ \frac{b}{2a}=+/-\sqrt{- \frac{c}{a} +( \frac{b}{2a} )^2}[/tex]

Step 6. Subtract [tex] \frac{b}{2a} [/tex] from both sides of the equation:
[tex]x+ \frac{b}{2a}- \frac{b}{2a} =- \frac{b}{2a} +/-\sqrt{- \frac{c}{a} +( \frac{b}{2a} )^2}[/tex]
[tex]x=- \frac{b}{2a} +/-\sqrt{- \frac{c}{a} +( \frac{b}{2a} )^2}[/tex]

Step 7. Simplify the radicand of the left hand side of the equation:
[tex]x=- \frac{b}{2a} +/-\sqrt{- \frac{c}{a} + \frac{b^2}{(2a)^2}} [/tex]
[tex]x=- \frac{b}{2a} +/-\sqrt{- \frac{c}{a} + \frac{b^2}{4a^2}} [/tex]
[tex]x=- \frac{b}{2a} +/-\sqrt{- \frac{4ac}{4a^2} + \frac{b^2}{4a^2}} [/tex]
[tex]x=- \frac{b}{2a} +/-\sqrt{- \frac{4ac+b^2}{4a^2} } [/tex]

Step 8. Take [tex]4a^2[/tex] outside the radical:
[tex]x=- \frac{b}{2a} +/- \frac{ \sqrt{-4ac+b^2} }{ \sqrt{4a^2} } [/tex]
[tex]x=- \frac{b}{2a} +/- \frac{ \sqrt{-4ac+b^2} }{2a} } [/tex]

Step 9. Combine the two fractions and rearrange the terms in the radical: 
[tex]x= \frac{-b+/- \sqrt{b^2-4ac} }{2a} [/tex]