Finding values of k for different points of intersection
My question is if you have a question like:
Find the values of K for which $y=kx-3$ does not intersect $y=x^2-2x+1$
To solve it my textbook says to do this.
$x^2-2x+1=kx-3$
$x^2-(2+k)x+4=0$
Then we use the discriminant.
$(2+k)^2-4\times 1\times 4<0$
$k^2+4k-12<0$
$(k+6)(k-2)<0$
So -6 < k < 2
My question is why does this work? What is actually happening for this to work? I've used Desmos and I can't figure out why the -6 < k < 2 part is equal to the resulting equation $k^2+4k-12<0$ between the two points of intersection on the x-axis.
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