how about this METHOD!:
you put the two equations together and then find the discriminate. if it is zero, then the tangent forms a "double root" with that parabola!
x =2y+2
x^2 =4y^2+8y+4
16y=4y^2+8y+4
0=4y^2-8y+4
0=y^2-2y+1
\Delta =b^2-4ac
\Delta =4-4(1x1)=0$ $ $Hence, only one solution...