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).

Fig. 1

Let’s find .

Line intersects at only means quadratic equation

has only one solution. That is, the discriminant of is zero:

Fig. 2

And, by the definition of slope (see Fig. 2),

.

It follows that

Substituting (2) into (1), we have

.

Solve it for gives

.

Fig. 3

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