When is this line a tangent to this parabola?

When is this line a tangent to this parabola?

Question

Calculate the value of $k$ for which the line $y = 2x + k$ is a tangent to the curve $y = 2x^2 - 4x + 5$.

Suggestion

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The blue parabola here is $y = 2x^2 - 4x + 5$, while the red line is $y = 2x + k$.

Solution

Note that $y=2x+k$ is a line, while $y=2x^2 -4x + 5$ is a parabola, so they meet in $0$, $1$ or $2$ points.

If $y=2x+k$ is a tangent to $y=2x^2 -4x + 5$, then they meet in exactly one point, so when we solve the two equations simultaneously, we must get just one solution.

Let’s do this: putting $2x+k = 2x^2-4x+5$ gives $2x^2 -6x + 5-k = 0$.

For this quadratic equation to have exactly one root, the discriminant $b^2-4ac$ must be zero. So $(-6)^2-4\times 2(5-k) = 0$, which gives $k = \dfrac{1}{2}$.