X^2-b(x-1)-1=0 find. (x)

x^2 - b x + (b - 1) = 0

x = [ b +/- sqrt (b^2 - 4b+4 ) ]/2

just use the quadratic formula:

x^2 - bx + b-1 = 0
x = [b±√(b^2-4(b-1))]/2
= [b±√(b^2-4b+1)]/2

If you want an integer solution for b, then clearly b=2 and you have

x^2-2x+1 = (x-1)^2 = 0

For a real solution, you need
b^2-4b+1 >= 0
or b >= 2+√3

Not sure just where you want to go with this.

Hmmm. I seem to have disregarded the distributive property ...

To find the value of x in the equation x^2 - b(x-1) - 1 = 0, we need to solve the quadratic equation. Let's go step by step.

Step 1: Expand the equation
x^2 - bx + b - 1 - 1 = 0
x^2 - bx + b - 2 = 0

Step 2: Rearrange the equation
x^2 - bx = 2 - b
x(x - b) = 2 - b

Step 3: Solve for x by dividing both sides by (x - b)
x = (2 - b)/(x - b)

So, the value of x in terms of b is x = (2 - b)/(x - b).