solving quadratic equations?

Question

Solve: x^3 +2x^2 +x > or = 0

 

Step by step assistance would be amazing!

Answer 1

Actually, your equation

 

x³+2x²+x≥0

 

is a cubic equation, since the highest-order term is x³. Fortunately, we can easily factor out an overall x,

 

x(x²+2x+1) ≥0,

 

and write the quadratic term as a complete square:

 

x(x+1)² ≥0.

 

How can the left-hand side be non-negative (which is what "≥0" means)?

 

Well, the term (x+1)² is always non-negative, because it's a square. In fact it is positive for all x≠-1, only for x=-1 is it zero.

 

So the entire left-hand side of the inequality is non-negative when x is non-negative, i.e. for all x≥0, and when x=-1. In interval notation, you could write the solution set as

 

{-1}∪[0, ∞).