CPM Homework Help : APCALC Problem 7-58

What is the value of $a$ such that the parabola $y = ax^2 + 3$ is tangent to the line $y = 5 - x$ ? Also, state the point of tangency.

Hint 1:

At the point of tangency, $ax^2 + 3 = 5 − x.$

Hint 2:

Since the tangent line is $y = 5 − x$ , then the slope of the tangent is $−1$ at the point of tangency.
Therefore $2ax = −1$ .

Hint 3:

Solve a system of equations.