Factor By Completing the Square

Question

x^2 + 21x -84 = 0

x^2 +21x = 84

x^2 +21x +441/4 = 777/4

I am able to get to this point....but how do I arrive at:

(x +21/2) = 777/4 <<<<<<< My question is NOT to solve, but how this line was derived.

Answer 1

when you got to

x^2 + 21 + 441/4 = 777/4

factor left side to get

(x+ 21/2)^2 = 777/4

as left side = (x+21/2)(x+21/2) = (using FOIL) = x^2 + 21x/2 + 21x/2 + 441/4 = x^2 +21x/2 + 441/4

x^2 +21x-84=0

take half the x term's coefficient and square it, then add and subtract it

x^2 + 21x + 441/4 - 84 - 441/4 = 0

(x+21/2)^2 - 336/4 - 441/4 = 0

(x+21/2)^2 = 777/4

take square roots of both sides

x+21/2 = +/-(1/2)sqr777

x = -10.5 +/-.5sqr777

= about -24.4373599.. or 3.4373599...

= about -24.4 or +3.4

check by graphing or use the quadratic formula or factor if possible

x = -21/2 +/- (1/2)sqr(441 +4(84))

= -10.5 +/-.5sqr777

= same answer

graph the parabola and find the x intercepts which = the same two answers