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

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

Step by step assistance would be amazing!

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New Wilmington, PA

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, ∞).

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