Question 250263
Let's start with original equation:
{{{x^2 + bx + c}}}
We set it equal to 0 and complete the square.
rewrite the expression as:
{{{x^2 + bx + _____ + c  = 0 +  _____}}}
The blanks are waiting for certain numbers.
step 1 - take 1/2 middle term and square it. Place that answer in the first blank and the opposite in the second blank as:
{{{x^2 + bx + (b/2)^2 + c = 0 + (b/2)^2 }}}
step 2 - subtract c from both sides to get
{{{x^2 + bx + (b/2)^2 = -c + (b/2)^2 }}}
step 3 - express the left side as a quantity squared as
{{{(x + (b/2))^2 = -c + (b/2)^2 }}}
Step 4 - simplify the right side as
{{{(x + (b/2))^2 = (b^2-4c)/4 }}}
step 5 - take a square root of both sides to get
{{{x + (b/2) = +- sqrt((b^2-4c)/4) }}}
step 6 - add b/2 to both sides to get
{{{x = -b/2 +- (sqrt(b^2-4c))/2 }}}
We now have our 2 roots as
r1 = {{{x = -b/2 + sqrt((b^2-4c))/2 }}}
and 
r2 = {{{x = -b/2 - sqrt((-b^2-4c))/2 }}}.
r1 + r2 = -b in this case, but in general it will be -b/2a.
r1*r2 = c in this case and in general it is c.