Question 713018

<pre>
{{{mx^2 + nx + o = 0}}}

{{{mx^2 + nx = -o

{{{expr(m/m)x^2 + expr(n/m)x = -o/m}}}


{{{x^2 + expr(n/m)x = -o/m}}}

Complete the square:

1. Multiply {{{expr(n/m)}}} by {{{1/2}}}:

    {{{expr(n/m)*expr(1/2)}}}
    {{{n/(2m)}}}

2. Square that result:

    {{{(n/(2m))^2}}}
     
    {{{n^2/(2^2m^2)}}}

    {{{n^2/(4m^2)}}}

3. Add to both sides of {{{x^2 + expr(n/m)x = -o/m}}}

{{{x^2 + expr(n/m)x + n^2/(4m^2) = -o/m + n^2/(4m^2)}}} 

Factor the left side;  Get LCD of {{{4m^2}}} on the right
and multiply the first term on the right by {{{(4m)/(4m)}}}

{{{(x + n/(2m))(x +n/(2m)) = expr(-o/m)expr(4m/(4m)) + n^2/(4m^2)}}}

{{{(x + n^2/(2m))^2 = (-4mo)/(4m^2) + n^2/(4m^2)}}}

{{{(x + n^2/(2m))^2 = (-4mo+ n^2)/(4m^2)}}}

Use the principle of square roots:

{{{x + n^2/(2m) =  "" +- sqrt( (-4mo+ n^2)/(4m^2))}}}

{{{x + n/(2m) =  "" +- sqrt( -4mo+ n^2)/sqrt(4m^2)}}}

 {{{x + n/(2m) =  "" +- sqrt( -4mo+ n^2)/(2m)}}}

    {{{ x = - n/(2m) +- sqrt( -4mo+ n^2)/(2m)}}}

    {{{x = (-n +- sqrt( -4mo+n^2 ))/(2m) }}} 

    {{{x = (-n +- sqrt(n^2-4mo ))/(2m) }}}
Edwin</pre>