Subfltitiite  in  ii    then      x'-+{6-x)'  =  18  ;

i.e.  2i'-12a:436-18,

Le.          a^«-6a>+9=    0,

(iii)

i.e.    *  -  3  or  8.
Sabatitateinii              y-3or3.
Hence  the  two  curves  meet  twice  at  the  point  (3,  S).
Fia.  80.     Soak :  8  aide*  -  1  unit.
214  QUADRATIC  EQUATIONS

The  two  curves  are  shown  in  Fig.  30,  where  the  circle  is  the
graph  of  a:^+y's  18  and  the  strsught  line  AB  is  the  graph  of
a;-f  y  B  6.  It  is  seen  that  the  line  ^B  just  meets  the  circle  at  the.
point  C  where  a;  «  3,  y  «=  3,  hut  it  does  not  cut  the  circle.  When
two  curves  or  lines  meet  hut  do  not  cut  they  are  said  to  touch.
This  is  sometimes  expressed  thus  :

Two  lines^ .  whether  straight  or  curved,  which  meet  in
two  coincident  points  are  said  to  touch  one  another.  Hence
we  see  that  since  the  equation  iii  has  two  equal  roots,  the
graphs  of  equations  i  and  ii  toucli  one  another.
Examples  XVII  o.
If  a  and  /3  are  the  roots  of  a^+px+q  «=  0,  find  the  values  of:
1 .  Ca  -  ^.X  V) -  2.  a«  - iS*.  3.  a»  -  ^\  4.  a«  +  P».
Without  actually  solving,  find  (i)  the  sum  of  the  roots,  (ii)  the
difference  of  the  roots,  (iii)  the  sum  of  the  squares  of  the  roots
of  each  of  the  following  equations :

5.  rc«-7ar  +  8  =  0.  6.  3a;«-4ir+16  =  0.
7.  2x^-Sx-b  =  0.  8.  ar«  +  21a?+6  =  0.
9.   ar»-2aaT-f  c  =  0.  10.  ir'  +  2a;r  +  a' =  7A

Form  the  simplest  equations  whose  roots  are
11.  2  +  ^5,  2-v^.  12.  3  and  -8.
13.  3-v'^,  3  +  -/^.        14.   -2+^13,  -2-yi3.
