Determine the value of k in y = -0.5x^2 - kx + 2 that will result in the intersection of the line y = -3x + 4 with the quadratic at one point ?

1 Answer
Apr 13, 2018

k = 5 or 1

Explanation:

We need the line to be tangent to the parabola. Since tangency is determined by the derivative, let's differentiate!

y' = -x - k

Since y' is the slope of the line, we get a third equation to add to the system.

-3 = -x - k

3 = x + k

Now we have

{(y = 4 -3x), (x+ k = 3), (y = -1/2x^2 - kx + 2):}

Substituting number three into number 1:

-1/2x^2 - kx + 2 = 4 - 3x

Now substitute 2 into 1.

-1/2x^2 -(3 -x)x + 2 = 4 - 3x

-1/2x^2 -(3x- x^2) + 2 = 4 - 3x

-1/2x^2 - 3x + x^2 = 2 - 3x

1/2x^2 = 2

x^2 = 4

x = +-2

Therefore, k = 3 - (-2) = 5 or k = 3 - 2 = 1

Hopefully this helps!