Q: For what value(s) of the constant will the curve y = x^3 + kx^2 + 3x - 4 have exactly one horizontal tangent?

I know:

Step 1: take a derivative:

y'=3x^2+2kx+3

Step 2: set it equal to 0 and solve for x

This is where I get stuck. This does not factor nicely, so we have to use the

quadratic formula. I got a little help from a classmate, but I'm not understanding the "WHY" part. Here is what I have, we have to use just the discriminant part (b^2-4ac) to find our answer. I know the answer is k=-3, 3 and I can complete the math part, it is just the understanding part that I'm lost on. Why wouldn't we use the whole quadratic formula? How is it legal to just use part of it?

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