1. Intersection of Functions

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.

2. Originally Posted by kaiser0792
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.
calculate the equation for

$\displaystyle f(x) = g(x)$
so,
$\displaystyle \frac{1}{x}$ = $\displaystyle \frac{{-x^2}+4x-2}{1}$

3. I realize that I need to set f(x) equal to g(x), but I'm not able to solve the resulting equation for x.

4. $\displaystyle \frac{1}{x} = -x^2 + 4x - 2$

$\displaystyle 1 = -x^3 + 4x^2 - 2x$

$\displaystyle 0 = -x^3 + 4x^2 - 2x - 1$

$\displaystyle 0 = x^3 - 4x^2 + 2x + 1$

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.

5. Thanks, I forgot about dividing out the known factor.