Find set of values for $k$ with two distinct roots?

Question

Find the set of values for $k$ for which the line $y=2x-k$ meets the curve $y=x^2+kx-2$ at two distinct points.

I have started my equation like this:

$$x^2+kx-2=2x-k$$

but I need to put it in the form $ax^2+bx+c$ (quadratic)

I know you have to use the discriminant $b^2-4ac > 0$ from there on, so really what I need help with is the first part of the question and rearranging the formulas.

Answer 1

Rearranging shouldn't be too hard; you have:

$$x^2+kx-2=\color{blue}{2x-k}$$

Get all terms to the same side by subtracting $\color{blue}{2x-k}$ from both sides:

$$x^2+kx-2\color{blue}{-2x+k}=0$$

Group per power of $x$:

$$x^2+(k-2)x+k-2=0$$

So now you can just read off the coefficients $a$, $b$ and $c$:

$$\underbrace{1}_{a}x^2+\underbrace{(k-2)}_{b}x+\underbrace{k-2}_{c}=0$$

Then proceed like you suggested (discriminant strictly positive).