Sign of a Quadratic Function with Application to Inequalities
Sign of a Quadratic Function with Application to Inequalities
Many inequalities lead to finding the sign of a quadratic expression. let us discuss this problem here. Consider the quadratic function
We know that
(double root case), then we have
In this case, the function 
has the sign of the coefficient a.
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(two distinct real roots case). In this case, we have
where 
and 
are the two roots with 
. Since 
is always positive when 
and 
, and always negative when 
, we get
has same sign as the coefficient a when and ; has opposite sign as the coefficient a when .
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(complex roots case), then has a constant sign same as the coefficient a.
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Example: Solve the inequality
Solution. First let us find the root of the quadratic equation 
. The quadratic formula gives
which yields x= -1 or x=2. Therefore, the expression
is negative or equal to 0 when 
.
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Author: Mohamed Amine Khamsi