Help Please!! (gr. 12 functions quick question)

Question

Solve the inequality

x³-2x²+2x-12 ≤ 0

I tried to solve it, but it doesn't factor any further. I don't know how to do it.

Answer 1

By sketching a graph of this cubic, we can see it has only one real root, x=2.809 approx. The solution is x≥2.809. Here's one way to find it. 

Let y=x³-2x²+2x-12. Now put x=0 so y=-12.

Next, x=1: y=-11; x=2: y=-8; x=3: y=27-18+6-12=3. So there is a change of sign between x=2 and x=3. Somewhere in this interval y=0 (curve crosses the x-axis). So we take a value between 2 and 3 for x, say 2.5. y<0 when x=2.5. That means we have a zero between 2.5 and 3, so we take another value between 2.5 and 3, and so on.

A quicker method is to use Newton's (iterative) Method: x_n+1=x_n-y(x_n)/y'(x_n), where y'(x)=3x²-4x+2.

Let starting value x₀=2, then x₁=10/3, x₂=662/297, x₃=2.812..., x₄=2.8089..., x=2.8089030726 approx.

So x≥2.8089 is an approximate solution.