A tangent line can be thought of as a line that "intersects
a curve twice at the same point".

The value of y when x = 2 is






So we want the equation of a tangent line at the point (2,10)

That is, we want to find the equation of the line through (2,10)
that is tangent to the parabola whose is equation is 




Let the tangent line have the equation 



The given equation of the parabola is



We set the two expressions for y equal to each other:





That quadratic will have a double solution if its
discriminant is 0

The discriminant is .  We use capital
letters to distinguish little b and capital B.





Solve that for b






Substitute that for b in





And also the point (x,y) = (2,10)





Multiply through by 4 to clear fraction:













Substitute this and (x,y) = (2,10) in









So the equation of the line is 



So we draw that line on the graph to see if it looks like what we
have calculated. It has y-intercept and slope -1, and goes through
points (-1,13), (0,12), (4,8)



Edwin