The equation of common tangent at the point of contact of two parabola $y^2=x$ and $2y=2x^2-5x+1$ is

$y^2=x\mathrm{........}\left(i\right)$

$2y=2x^2-5x+\mathrm{1.......}\left(ii\right)$

Solving (i) & (ii) ${(y+1)}^2(2y^2-4y+1)=0$

$y=-1$ is the repeated root which is the y-(o-ordinate of point of contact. So point of contact =(1,-1)

Common tangent at point of contact $y(-1)=\frac{x+1}{2}\Rightarrow x+2y+1=0$