What are the equations of the tangent lines to both functions?

[teken894]

Here's a question from my Calc HW andI believe my approach is flawed...

Two functions
f(x) = x^2
g(x) = -x^2 + 6x -5

Find the two lines tangent to both functions


I thought I could find the slope where f' and g' are equal so I did:

f'(x) = 2x
g'(x) = -2x + 6
2x = -2x +6
x = 1.5
slope = 2(1.5) = 3

BUT that slope doesn't work as this simply finds an x on the two graphs where the two graphs have the same slope.


Help me please!

THE given answers are
y=2x-1
y=4x-4

 

[Fermat]

Assume the line of tangency is y = mx +c.

Solve to find the point of intersection of y=mx + c with each of the two curves.

Keep in mind that there is only one point of intersection (with each curve) and not two x-values.