How do you complete the square for #x^2+18x#?

2 Answers
May 31, 2015

#(x+9)^2 = x^2 + 18x +81#

In general,

#ax^2+bx+c = a(x+b/(2a))^2 + (c - b^2/(4a))#

Notice that the term added to #x# is #b/(2a)#

May 31, 2015

For a general form, squared binomial
#color(white)("XXXXX")##(x+a)^2 = x^2+2ax+a^2#

So if #x^2+18x# are the first two terms of a squared binomial
#color(white)("XXXXX")#then, in the general form, #a=9# and
#color(white)("XXXXX")##a^2 = 9^2 = 81#

Of course, if we are going to add #9^2# to the expression #x^2+18x# we are also going to have to subtract it:
#color(white)("XXXXX")##x^2+18x#
#color(white)("XXXXX")##= x^2+18xcolor(red)(+9^2) - color(blue)(9^2)#
#color(white)("XXXXX")##=color(red)((x+9)^2) color(blue)(- 81)#