Find the value of $k$ if the curve $y = x^2 – 2x$ is tangent to the line $y = 4x + k$
I have looked at the solution to this question and the first step is the "equate the two functions":
$ x^2 – 2x = 4x + k$
Why? How does that help solve the equation? And how can I use what I get from equating the two functions to find the solution?
Best Answer
Additionally, you can also solve with calculus. Take the derivative. $(x^2-2x)' = 2x-2$. We want the derivative to be $4$, which is the slope of the line. This happens when $2x-2 = 4$, at $x=3$.
Plugging into our quadratic: $y=3^2-2\cdot 3 = 3$.
So we need $y = 4\cdot 3 + k$, or $3 = 4\cdot 3 + k$. This gives $\boxed{k=-9}$.