Many inequalities lead to finding the sign of a quadratic expression. let us discuss this problem here. Consider the quadratic function
We know that
In this case, the function has the sign of the coefficient a.
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where and
are the two roots with
. Since
is always positive when
and
, and always negative when
, we get
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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
.