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

displaymath21

We know that

1
if tex2html_wrap_inline23 (double root case), then we have

displaymath25

In this case, the function tex2html_wrap_inline27 has the sign of the coefficient a.

a<0
a>0

2
If tex2html_wrap_inline31 (two distinct real roots case). In this case, we have

displaymath33

where tex2html_wrap_inline35 and tex2html_wrap_inline37 are the two roots with tex2html_wrap_inline39 . Since tex2html_wrap_inline41 is always positive when tex2html_wrap_inline43 and tex2html_wrap_inline45 , and always negative when tex2html_wrap_inline47 , we get

a<0
a>0

3
If tex2html_wrap_inline63 (complex roots case), then tex2html_wrap_inline49 has a constant sign same as the coefficient a.

a<0
a>0

Example: Solve the inequality

displaymath69

Solution. First let us find the root of the quadratic equation tex2html_wrap_inline71 . The quadratic formula gives

displaymath73

which yields x= -1 or x=2. Therefore, the expression tex2html_wrap_inline79 is negative or equal to 0 when tex2html_wrap_inline81 .

[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

Copyright 1999-2017 MathMedics, LLC. All rights reserved.
Contact us
Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA
users online during the last hour