the distance between (4,1) and (x,3) is the square root of 52. find the values of x

Let's make this easy on ourselves....basically, we need to solve this

(4 - x) ^2 + (3 - 1)^2  = 52     simplify

(4 - x)^2  + (2) ^2  = 52

(4 - x)^2  + 4  = 52     subtract 4 from both sisdes

(4 - x)^2  = 48           

(x -  4) ^2   = 48     take both roots

x - 4  = ±√48          add 4 to both sides

x = ±√48 + 4  =  ±4√ 3 + 4  = 4( 1±√ 3)

So...two "x's" make this true...