Question 532001
3x^2 - 2x + __ = 0
to complete the square we need the coefficient of x^2 to 1, divide eq by 3,
x^2 - {{{2/3}}}x + ____ = 0
to obtain the term that completes the square, divide the coefficient of x by 2 and square it.
 Add of both sides
x^2 - {{{2/3}}}x + {{{1/9}}} = {{{1/9}}}
A perfect squares which factors to
{{{(x - 1/3)^2}}} = {{{1/9}}}
find the square root of both sides
{{{(x - 1/3)}}} = +/-{{{sqrt(1/9)}}}
Find the square root of 1/9
x = {{{1/3}}} +/-{{{1/3}}}
Two solutions
x = {{{1/3}}} + {{{1/3}}}
x = {{{2/3}}}
and 
x = {{{1/3}}} - {{{1/3}}}
x = 0
:
:
 x^2 + 100 = 27x 
Write it in this form
 x^2 - 27x + ___ = -100
Find the 3rd term that completes the square, (27/2)^2
 x^2 - 27x + {{{729/4}}} = -100 + {{{729/4}}}
 x^2 - 27x + {{{729/4}}} = {{{-400/4}}} + {{{729/4}}}
 x^2 - 27x + {{{729/4}}} = {{{329/4}}}
 {{{(x - 27/2)^2}}} = {{{329/4}}}
Find the square root of both sides
 {{{(x - 27/2)}}} = +/- {{{sqrt(329/4)}}}
x = {{{27/2)}}} +/- {{{sqrt(329/4)}}}
Extract the square root of 1/4
x = {{{27/2)}}} +/- {{{1/2}}}*{{{sqrt(329)}}}
Two solutions
x = {{{(27 + sqrt(329))/2}}}
and
x = {{{(27 - sqrt(329))/2}}}