Find an equation for all lines which pass through the point (2,3) and are tangent to the parabola y=x^2
This is a trick question because the point (2,3) is not on the parabola.
the slope of any such line must be $\displaystyle m=2x$ and of the form ...
$\displaystyle y - 3 = 2x(x - 2)$
simplify ...
$\displaystyle y = 2x^2 - 4x + 3$
since each tangent line also touches the the parabola ...
$\displaystyle 2x^2 - 4x + 3 = x^2$
solve for x, then get the equations of the two lines that meet the conditions set forth in the problem.