Answer:
Explanation:
To find the value of x^2 + b^2, we can start by solving the given equations.
First, let's rearrange the equation x + b = 2 to solve for x:
x = 2 - b
Now, let's calculate the distance between the points (1, x) and (b, 1) using the distance formula:
Distance = sqrt((x - 1)^2 + (b - 1)^2)
Since the distance is given as 3, we can substitute the values into the equation:
3 = sqrt((x - 1)^2 + (b - 1)^2)
Next, let's square both sides of the equation to eliminate the square root:
9 = (x - 1)^2 + (b - 1)^2
Expanding the equation, we have:
9 = x^2 - 2x + 1 + b^2 - 2b + 1
Simplifying, we get:
9 = x^2 + b^2 - 2x - 2b + 2
Now, let's substitute x = 2 - b into the equation:
9 = (2 - b)^2 + b^2 - 2(2 - b) - 2b + 2
Expanding and simplifying further:
9 = 4 - 4b + b^2 + b^2 - 4 + 2b - 2b + 2
Combining like terms, we get:
9 = 2b^2 - 2b + 2
To find the value of x^2 + b^2, we need to substitute the value of x:
x^2 + b^2 = (2 - b)^2 + b^2
Expanding and simplifying:
x^2 + b^2 = 4 - 4b + b^2 + b^2
Combining like terms, we have:
x^2 + b^2 = 2b^2 - 4b + 4
Therefore, the value of x^2 + b^2 is 2b^2 - 4b + 4.