# 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 . [Algebra] [Complex Variables]
[Geometry] [Trigonometry ]
[Calculus] [Differential Equations] [Matrix Algebra] S.O.S MATHematics home page

Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard. Author: Mohamed Amine Khamsi