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Quadratic Inequalities - Using Factorization, Formulas and Plots of Curves- A Tutorial with Solved Problems and a Quiz at the end


Target Audience: High School Students, College Freshmen and Sophomores, Class 11/12 Students in India preparing for ISC/CBSE and Entrance Examinations like the IIT-JEE 

Main or Advanced/AIEEE

, and anyone else who needs this Tutorial as a reference!

After understanding this topic, you might benefit from the MCQ Quiz over here.



 

Quadratic Inequalities

Quick Introduction (more detailed introduction in the tutorial document)

Quadratic inequalities refer to the inequalities of the type: ax22

The basic concepts in inequalities are:
  • if ab>0 then either a>0 and b>0 or a<0 and b<0.
  • if ab<0 then either a>0 and b<0 or a<0 and b>0.
Using this, if we have factorized the expression in the form: (x-p)(x-q),and p<q, then : if (x-p)(x-q)>0 then either x-p>0 and x-q>0 or x-p<0 and x-q<0
=>x>p and x>q or x<p and x<q
=>x>q or x<p (Because if x>p and x>q, then x>q is the common solution as p<q(Notice that “and”
leads to taking the common solution, and “or” leads to unifying the solutions) )
Similarly, if (x-p)(x-q)<0 then, p<x<q.
Clearly, when x<α and when x>β, we have a positive value of f(x), where f(x)= ax2+bx+c when α<x< β, we have a negative value of f(x)

Tutorial with solved problems (To test your knowledge, try out the quiz at the end):





MCQ Quiz/Worksheet for Quadratic In-equations

Your score will be e-mailed to you at the address you provide.

MCQ Companion Quiz- Quadratic Inequalities




 Introduction to Complex Numbers
Introduction to Complex Numbers and iota. Argand plane and iota. Complex numbers as free vectors. N-th roots of a complex number. Notes, formulas and solved problems related to these sub-topics.		 The Principle of Mathematical InductionIntroductory problems related to Mathematical Induction.Quadratic Equations
Introducing various techniques by which quadratic equations can be solved - factorization, direct formula. Relationship between roots of a quadratic equation.  Cubic and higher order equations - relationship between roots and coefficients for these. Graphs and plots of quadratic equations.		Quadratic Inequalities
 Quadratic inequalities. Using factorization and visualization based methods.
 Series and Progressions
Arithmetic, Geometric, Harmonic and mixed progressions. Notes, formulas and solved problems. Sum of the first N terms. Arithmetic, Geometric and Harmonic means and the relationship between them.		 


 

After reading this tutorial you might want to check out some of our other Mathematics Quizzes as well.
 Quizzes on Progressions
MCQ #1: Arithmetic Progression 
MCQ #2: Geometric Progression
MCQ #3 : More on Geometric Progressions.
MCQ #4 : Harmonic Progressions. 
MCQ #5: More on Harmonic Progression
MCQ #6: Mixed Progressions


Quadratic Equations
MCQ Quadratic Equations

Quadratic In-equations
MCQ Quadratic In-equations
 Coordinate Geometry - Straight Lines
MCQ #1: Cartesian Planes, Straight Line Basics
MCQ #2 on Straight Lines
MCQ #3 on Straight Lines
MCQ #4 on Straight Lines

Circles
1 MCQ #1 on Circles. 
2 MCQ #2 on Circles. 
3 MCQ #3 on Circles. 

Conic Sections- Parabola, Hyperbola, Ellipse
1 MCQ- The Basics of Conic Sections
2 MCQ on Parabola..
3 MCQ on Hyperbola
4 MCQ on Ellipses. 		 Probability
MCQ #1 on Basic Probability
MCQ #2: More Challenging Problems on Probability
MCQ #3- Conditional Probability and Bayes Theorem