Answer
For the parabola to be tangent to the line, $C$ needs to equal $1/4$.
Work Step by Step
The parabola $y=x^2+C$ and the line $y=x$.
When it comes to the relationship between a curve and a line, if they intersect at only one point, we can conclude that the line is a tangent to the curve.
So here, we need to find $C$ so that the line intersects the parabola at only one point, or, the equation $x^2+C=x$ has only one solution.
$$x^2+C=x$$ $$x^2-x+C=0$$
Recall that for a quadratic equation $y=ax^2+bx+c$, we have $$\Delta=b^2-4ac$$
If we want the equation to have only one solution, then $\Delta$ needs to equal $0$.
In this case, for the equation $x^2-x+C=0$, we have $a=1$, $b=-1$ and $c=C$.
$$\Delta=(-1)^2-4\times1\times C=1-4C$$
Then for $\Delta=0$, $$1-4C=0$$ $$C=\frac{1}{4}$$
Therefore, for the parabola to be tangent to the line, $C$ needs to equal $1/4$.
