The Learning Point‎ > ‎Mathematics‎ > ‎

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: ax2+bx+c>0 or ax2+bx+c<0  (or with the inequalities with the equal to sign)

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)

Here are the kind of questions covered in the tutorial

Q: Solve x2+x+1<0

Q:  Solve x2+4x>0

Q: Solve x2+5x+1>0.

Q: Solve x2+11x+24<0

Q: If x2+kx+2=0 has real roots, what can be the possible values of k?

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

You might like to take a look at our other algebra tutorials:

 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 Induction Introductory 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.