Question 134607
{{{x-9=sqrt(x-3)}}}


Square both sides:


{{{x^2-18x+81=x-3}}}


Add -x+3 to both sides:


{{{x^2-19x+84=0}}}


Factor:
{{{-7*-12=84}}} and {{{-7-12=-19}}}, so:


{{{(x-7)(x-12)=0}}}


{{{x=7}}} or {{{x=12}}}


Check:


{{{7-9=sqrt(7-3)}}}
{{{-2<>2}}}, therefore 7 is an extraneous root introduced by squaring both sides of the equation at the start.


Exclude 7.


Why not say {{{sqrt(4)=-2}}}?  Because by convention {{{sqrt(x)}}} is the positive root.  {{{-sqrt(4)=-2}}}


{{{12-9=sqrt(12-3)}}}
{{{3=sqrt(9)=3}}}


The solution set is {12}.