if disatance between two points (a+2,3)and (5,-1) is 5 units,find the value of a.

For convenience we work with d^2 instead of d 


 

becomes: 
 


then 

 
Notice that after we square a real quantity the result is positive so
(3-a)^2=(a-3)^2 
and so we rewrite it as this: 


 


 


 

this means a-3=3 or a-3=-3 

and so a=6 or a=0 

checking... 
(5-(0+2))^2+(-1-3)^2=3^2+4^2 
and so 0 works 

(5-(6+2))^2+(-1-3)^2=(-3)^2+(-4)^2 
and so does 6 


:)