Question 4159
{{{a*x^2+b*x+c=0}}}
by complete square:
add and subtract {{{b^2/(4*a)}}}
{{{a*x^2+b*x+c+(b^2/(4*a))-(b^2/(4*a))=0}}}
Rewrite the equation
{{{(a*x^2+b*x+b^2/(4*a))-b^2/(4*a)+c=0}}}
Factor for a
{{{a*(x^2+b/a*x+b^2/(4*a^2))-b^2/(4*a)+c=0}}}
Factor the complete square {{{(x^2+b/a*x+b^2/(4*a^2))}}}
{{{(x^2+b/a*x+b^2/(4*a^2))=(x+b/(2*a))^2}}}
{{{a*(x+b/(2*a))^2-b^2/(4*a)+c=0}}}
{{{a*(x+b/(2*a))^2=b^2/(4*a)-c}}}
{{{a*(x+b/(2*a))^2=(b^2-4*a*c)/(4*a)}}}
Multiply both sides by {{{1/a}}}
{{{(x+(b/(2*a)))^2=(b^2-4*a*c)/(4*a^2)}}}
Take the square root for both sides
{{{x+b/(2*a)=sqrt((b^2-4*a*c)/(4*a^2))}}} or {{{x+b/(2*a)=-sqrt((b^2-4*a*c)/(4*a^2))}}}
{{{x=(-b)/(2*a)+sqrt(b^2-4*a*c)/(2*a)}}} or {{{x=(-b)/(2*a)-sqrt(b^2-4*a*c)/(2*a)}}}
{{{x=(-b+-sqrt(b^2-4*a*c))/(2*a)}}}