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!
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.
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.
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. 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.