I need help finding the points at which the following 2 functions intersect:
f(x) = 1/x
g(x) = -x^2 + 4x - 2
for x>0
I figured out that x=1 was a solution, by inspection, but I cannot solve for the other one.
$\displaystyle \frac{1}{x} = -x^2 + 4x - 2$
$\displaystyle 1 = -x^3 + 4x^2 - 2x$
$\displaystyle 0 = -x^3 + 4x^2 - 2x - 1$
$\displaystyle 0 = x^3 - 4x^2 + 2x + 1$
Clearly we need to factor the right-hand side to find the solutions. You already know $\displaystyle (x-1)$ is a factor because you said that $\displaystyle x=1$ is a solution. So, use polynomial long division (or synthetic division) to factor and find the other solutions.