Hence  multiplying  the  equation  by  4  will  avoid  the  introduc-
tion of  fractions,  and  also  leave  the  square  of  the  whole  of  the
co-efficient  of  the  lowest  power  to  be  added  to  both  sides  ac-
cording to  the  rule.
The  first  term  evidently  continues  to  be  a  square  after  it  is
multiplied  by  4,  for  it  is  still  the  product  of  the  powers  of
certain  factors.  (Art.  167.)

3.  It  will  be  perceived  at  once,  that  the  second  term  is
composed  of  twice  the  root  of  the  first  term  into  the  co-effi-
cient of  the  last  term,  which  constitutes  the  middle  term  of  a
binomial  square.  (Art.  173.)

Obser.  It  is  manifest  from  the  preceding  demonstration,  that  multiply-
ing by  4  is  not  a  necessary  step  in  completing  the  square,  but  is  resorted  to
as  an  expedient  to  prevent  the  occurrence  of  fractions.  When  therefore
the  co-efficient  of  the  lowest  power  is  an  even  number,  so  that  half  of
it  can  be  taken  without  a  remainder,  we  may  simplify  the  operation
by  multiplying  by  the  co-efficient  of  the  highest  power  alone,  and  ad-
ding to  both  sides  the  square  of  half  the  co-efficient  of  the  lowest  power
of  the  unknown  quantity.
Take  the  equation,  7z2-f-40z=7lf-.
Multiplying  by  7,  it  becomes  49z2-f-280z=500

Adding  the  square  of  half  the  co-efficient,    49z2-f-280z-|-4 00=900
By  evolution  and  transposition,  x=40.
265.  From  the  preceding  principles  we  may  also  deduce

OTHER  METHODS  OF  COMPLETING  THE  SQUARE.
Multiply  the  equation  by  16  times  the  co-efficient  of  the
highest  power  of  the  unknown  quantity,  and  add  to  both  sides
4  times  the  square  of  the  co-efficient  of  the  lowest  power.
