The tangent line of a quadratic function at is a line that intersects at only.
The presence of function immediately excludes the vertical line which also intersects at only (see Fig. 1).
Let’s find .
Line intersects at only means quadratic equation
has only one solution. That is, the discriminant of is zero:
And, by the definition of slope (see Fig. 2),
It follows that
Substituting (2) into (1), we have
Solve it for gives