Question

The surface of a machine part is the region between the graphs of
y1 = |x| and y2 = 0.08x^2 + k

Answer 1

set curves equal to each other

.08x^2 +k = x

.08x^2 - x + k = 0

tangent means touching at 1 point so there must only be 1 solution
set discriminant equal to 0

b^2 - 4ac = 0

Now you can find "k"

Answer 2

So It would be -x^2-4(.08x^2)(k)=0 to find k?

Answer 3

The discriminant uses the coefficients of your quadratic equation. It is this part of the quadratic formula, if you're not familiar with it...
\( \displaystyle x = \frac{-b \pm \sqrt{\color{#aaaa00}{b^2 - 4ac}}}{2a} \)
The aspect of having only one solution comes down to that part equaling zero because you receive the solution x = (-b +- 0)/2a = -b/2a.

Answer 4

So from the general equation, \( ax^2 + bx + c = 0\)
\( 0.08 x^2 - x + k = 0 \)
Or to make it easier to see,
\(0.08 x^2 + (-1)x + k = 0\)