Math Homework

My calc homework confuses me, please help.

Question:
y= x^2
y= - x^2 + 2x - 5

Find the equations of two lines that are simultaneously tangent to both parabolas.

>>4
The x value such that the derivatives are equal does not (necessarily) give the slope of a line which is tangent to both curves. The lines are 4x - 4 and -2x - 1. You find this by expressing a tangent to f(t) = t^2 (the first function) as a function of t and x (specifically, y = 2tx - t^2), then subtracting that from the second function and using the quadratic formula to find which values of t result in a double root (ie tangent line), then substituting those values of t back into y = 2tx - t^2 to get the line equations.