Common tangent lines of two quadratic functions Find all such lines that are tangent to the following curves: $$y=x^2$$ and $$y=-x^2+2x-2$$ I have been pounding my head against the wall on this. I used the derivatives and assumed that their derivatives must be equal at those tangent point but could not figure out the equations. An explanation will be appreciated.

Here is a hint for a method which avoids calculus:

The line $y=ax+b$ is a tangent to a quadratic such as $y=x^2$ if and only if the quadratic equation you get by solving these equations simultaneously has a double root. This will give you an equation which must be satisfied by the unknowns $a$ and $b$.

You can do the same for the line $y=ax+b$ and your other quadratic, then solve the two simultaneous equations to find $a$ and $b$.