Vertical distance between two parabola

1. The problem statement, all variables and given/known data

Write a function for d(x), the vertical distance between the two curves, and find the minimum value of d(x).

2. Relevant equations

The equation for parabola one is y = x^2 + 6, for parabola two, y = -(x-2)^2 + 6

3. The attempt at a solution

The answer in the back of the book is d(x) = 2x^2 - 4x + 4, with a minimum value of 2. This is from my old algebra 2 trig textbook and I have no teacher to ask for help, as I am doing self study. Any and all help would be very much appreciated.
I figured it out, its just d(x) = y1 - y2, seems I was just over thinking things... but I have another question :)

Find the Value(s) of k for which the graph y = kx intersects the graph of y = x^2 + 25 in only one point.

If I set the equations equal, I get 0 = x^2 - kx + 25, and I know perfect squares have only one root, so it seems as though the answer is 10 or -10. Does this seem reasonable?


Science Advisor
Homework Helper
Yes, I think it does.
from 0 = x2 - kx + 25, use the discriminant method to find the value of k that will result to a single intercept only. that is

b2 - 4ac = 0 for one root or one intercept
b2 - 4ac > 0 for two roots or two intercepts
b2 - 4ac < 0 if you wish them to have no intercepts
oh i forgot...

a = 1
b = -k
c = 25
x2 = x squared

tnx tnx ^__^__^

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