The graph of the parabola $y = x^2 + 2bx + c$ is tangent to the line $y = 9x - 7$, where $c \neq 0$. What is $c-4b$?

Answer 1

Setting the two expressions for $y$ equal, we have $x^2 + 2bx + c = 9x - 7$, or \[x^2 + (2b - 9)x + (c + 7) = 0.\]Since the graph of the parabola is tangent to the line, the quadratic has exactly one root. This occurs if and only if the discriminant is equal to 0, so \[(2b - 9)^2 - 4(c + 7) = 0.\]Expanding, we get $4b^2 - 36b + 81 - 4c - 28 = 0$, or $4b^2 - 36b - 4c + 53 = 0$. Then $c = b^2 - 9b + 13$, so $c - 4b = b^2 - 9b + 13 - 4b = b^2 - 13b + 13 = \boxed{\frac{169}{4}}.$

Answer 2

This step is wrong:
4b^2 - 36b - 4c + 53 = 0
c = b^2 - 9b + 13
since 4*13 is not 53.