How do you find the equation of both lines through the point (2, -3) that are tangent to the parabola y=x^2-x? and then show that there is no tangent line through the point (2, 7)?

Any tangent line to at a point will have slope . Such a line has equation



For this (these) line(s) to pass through (2, -3), you must have



So the two lines in questions are for these values of .

To show there is no tangent line that passes through (2, 7), you have to show there is no solution for in



You have



but this has no real solutions, so no such tangent line exists.