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solving quadratic equations?

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1 Answer

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, ∞).
 
 

Comments

From the quadratic term, (x+1)². If x=-1, this term is zero, so it's a solution of your inequality!

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