Answer: To see if the expression 2x + (x - 7)^2 is equivalent to x^2 + bx + 49 for all values of x, we can expand the square in the first expression and simplify:
2x + (x - 7)^2 = 2x + (x^2 - 14x + 49)
= x^2 - 12x + 49
Now we can compare this with the second expression, x^2 + bx + 49. Since these two expressions are equivalent for all values of x, their coefficients must be equal. In other words:
x^2 - 12x + 49 = x^2 + bx + 49
Simplifying, we get:
-12x = bx
Dividing both sides by x (note that x cannot be zero because it is in the domain of both expressions), we get:
-12 = b
Therefore, the expression 2x + (x - 7)^2 is equivalent to x^2 - 12x + 49, which is equivalent to x^2 - 12x + 49 for all values of x. The coefficient of x in the second expression is -12, so b = -12.
Explanation: