SOLUTION: x^2 - 6x- (x^2 -6x - 3 )^1/2 =5

Since x²-6x appears twice, let u = x²-6x

Isolate the square root on the right side:

Square both sides


Get 0 on the right

Factor the left side:
(u-4)(u-7) = 0
u-4 = 0;  u = 7
  u = 4;  u = 7
Since u = x²-6x
  x²-6x = 4;       x²-6x = 7
x²-6x-4 = 0;     x²-6x-7 = 0
              (x-7)(x+1) = 0
               x-7 = 0; x+1 = 0
                 x = 7;   x = -1
Since x²-6x-4 does not factor 
                
    










Four potential solutions.

7, -1, , 

But we must check all solutions when there is a square
root radical with a variable underneath contained in the 
original equation.

Checking x=7 in the original equation:







That checks, So x=7 IS a solution

Checking x=-1 in the original equation:








That checks, So x=-1 IS a solution
              
Checking x=3 + sqrt(13) in the original equation:


That's too hard to substitute in so, do it on 
your TI-84:

Type in 
Press STO 
Press ALPHA STO ENTER
Type in 
Press ENTER
Read 3.

Since 3 is not 5, 3 + sqrt(13) is an EXTRANEOUS answer,
so we discard it.


Checking x=3 - sqrt(13) in the original equation:


That's also too hard to substitute in so, do it on 
your TI-84:

Type in 
Press STO 
Press ALPHA STO ENTER
Type in 
Press ENTER
Read 3.

Since 3 is not 5, 3 - sqrt(13) is also an EXTRANEOUS answer,
so we discard it.

The only solutions are 7 and -1.

Edwin

x^2 - 6x- (x^2 -6x - 3 )^1/2 =5 --->

(x^2 - 6x -3) - (x^2 -6x - 3 )^1/2 = 2.   (*)

Introduce new variable 

u = (x^2 -6x - 3 )^1/2.

Then the equation (*) takes the form

u^2 - u = 2,   or

u^2 - u - 2 = 0.

Factor:

(u-2)*(u+1) = 0.

The roots are u = 2  and  u = -1.

Now you have this equations for x:


     (x^2 -6x - 3 )^1/2 = 2  --->  x^2 -6x - 3 = 4  --->  x^2 -6x -7 = 0.

     Factor: (x-7)*(x+1) = 0  --->  the roots are x= 7 and x= -1.