Question 549677
<pre>
Start with the general quadratic equation:

 Ax² + Bx + C = 0

Subtract C from both sides

     Ax² + Bx = -C

Divide through by A

    {{{A/A}}}x² + {{{B/A}}}x = {{{-C/A}}}

Simplify:

     x² + {{{B/A}}}x = {{{-C/A}}}

                                   Multiply the coefficient of x by {{{1/2}}}:  {{{expr(B/A)*expr(1/2)}}} = {{{B/(2A)}}}
                                   Square {{{B/(2A)}}}.  {{{(B/(2A))^2}}} = {{{B^2/(4A^2)}}}
                                   Add {{{B^2/(4A^2)}}} to both sides:

x² + {{{B/A}}}x + {{{B^2/(4A^2)}}} = {{{B^2/(4A^2)}}} - {{{C/A}}}

Factor the left side and get LCD of 4A² on the right

(x + {{{B/(2A)}}})(x + {{{B/(2A)}}}) = {{{B^2/(4A^2)}}} - {{{C/A}}}{{{((4A)/(4A)))}}}

Write left side as the square of a binomial.  Multiply fractions on far right

(x + {{{B/(2A)}}})² = {{{B^2/(4A^2)}}} - {{{4AC/(4A^2)}}}

Combine fractions on the right over the LCD

(x + {{{B/(2A)}}})² = {{{(B^2-4AC)/(4A^2)}}} 

Use the principle of square roots:

x + {{{B/(2A)}}} = {{{"" +- sqrt((B^2-4AC)/(4A^2))}}}

Take square roots of top and bottom on the right:

x + {{{B/(2A)}}} = {{{"" +- sqrt(B^2-4AC)/sqrt(4A^2))}}}

x + {{{B/(2A)}}} = {{{"" +- sqrt(B^2-4AC)/(2A)}}}

Add {{{-B/(2A)}}} to both sides:

               x = {{{-B/(2A)}}} ± {{{sqrt(B^2-4AC)/(2A)}}}

Combine the fractions on the right over 2A

               x = {{{(-B +- sqrt(B^2-4AC))/(2A)}}}


Edwin</pre>