Given information:
- Curve: f(x) = x^2 - bx - b
- Point on the curve: (1, 1)
- Triangle is formed by the tangent to the curve at point (1, 1) and the coordinate axes
- Area of the triangle = 2

Find:
Value of b

Solution:
Step 1: Finding the equation of the tangent at point (1, 1)

To find the equation of the tangent at point (1, 1), we need to find the slope of the tangent. The slope of the tangent is equal to the derivative of the function at that point.

Differentiating the function f(x) = x^2 - bx - b with respect to x, we get:
f'(x) = 2x - b

Substituting x = 1 into the derivative, we get:
f'(1) = 2(1) - b = 2 - b

So, the slope of the tangent at point (1, 1) is 2 - b.

Now, we can find the equation of the tangent using the point-slope form:
y - y1 = m(x - x1), where (x1, y1) is the given point on the curve.

Substituting the values (1, 1) and m = 2 - b, we get:
y - 1 = (2 - b)(x - 1)

Simplifying the equation, we get:
y = (2 - b)x - (1 - b) (Equation 1)

Step 2: Finding the coordinates of the triangle vertices

The triangle is formed by the tangent and the coordinate axes. To find the coordinates of the vertices, we need to find the x-intercept and y-intercept of the tangent.

X-intercept:
To find the x-intercept, we set y = 0 in Equation 1:
0 = (2 - b)x - (1 - b)

Simplifying the equation, we get:
x = (1 - b)/(2 - b) (Equation 2)

Y-intercept:
To find the y-intercept, we set x = 0 in Equation 1:
y = (2 - b)(0) - (1 - b)

Simplifying the equation, we get:
y = b - 1 (Equation 3)

The coordinates of the vertices are (0, b - 1) and ((1 - b)/(2 - b), 0).

Step 3: Finding the area of the triangle

The area of the triangle can be found using the formula:
Area = (1/2) * base * height

In this case, the base of the triangle is the x-intercept and the height is the y-intercept.

Using Equations 2 and 3, we can substitute these values into the area formula:
Area = (1/2) * (1 - b)/(2 - b) * (b - 1)

Simplifying the equation, we get:
Area = (1/2) * (b - 1)/(2 - b)

Given that the area is 2, we have:
2 = (1/2)