Question 151721
What is the actual quadratic formula, and how would you actually get the equation itself? All formulas have a reason to how it was made. Such as, a stated theorem.
:
consider ax^2 = bx + c = 0
:
solve this by completing the square of the left side, we have:
{{{x^2 + (b/a)x + (c/a)}}} = 0
;
adding {{{-c/a}}} to both sides and completing the square on the left side:
 {{{x^2 + (b/a)x + (b^2/(4a^2))}}} = {{{(b^2/(4a^2)) - (c/a)}}}
:
Now if we factor the left side, using the square root property
{{{(x +(b/(2a)))^2}}} = {{{(b^2-4ac)/(4a^2)}}}
Finding the square root of both sides
{{{x +(b/(2a))}}} = +/-{{{sqrt((b^2-4ac)/(4a^2))}}}
:
x = {{{(-b/(2a))}}} +/- {{{sqrt(b^2-4ac)/(2a)}}}
We end up with the famous quadratic formula, that we all love so well!
x = {{{(-b +- sqrt( b^2-4*a*c ))/(2*a) }}}