Question 1055142
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x^2 - 6x- (x^2 -6x - 3 )^1/2 =5
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Below find more short solution.


<pre>
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.
</pre>


{{{graph( 330, 330, -10.5, 10.5, -10.5, 10.5,
          x^2 - 6x- (x^2 -6x - 3 )^1/2, 5
)}}}


Plot y = {{{x^2 - 6x- (x^2 -6x - 3 )^(1/2)}}} and y = 5