Finding the values of a and b in the function $f(x) = x^2 + ax + b$

Question

So I have to find the values of $a$ and $b$ in the function $f(x) = x^2 + ax + b$. What I have is that the function has the tangent line $2y + 4x = 2$ at the point $(2,5)$.

Not quite sure how to solve this.

Answer 1

The first condition is that $f(2)=5$, because it passes through $(2,5)$ which gives $$2a+b=1$$ The slope of the line $2y+4x=2$ is $-2$. Since $f(x)$ is tangent to it at $(2,5)$, we have $$ \begin{align} f'(2)&=-2\\ 4+a&=-2\\ a&=-6 \end{align} $$ Which gives $b=13$