# 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

1
if (double root case), then we have

In this case, the function has the sign of the coefficient a.

 a<0 a>0

2
If (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 .
 a<0 a>0

3
If (complex roots case), then has a constant sign same as the coefficient a.

 a<0 a>0

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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