Question

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 ?

Answer 1

`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!