JEE Quadratic Roots
This page will teach you how to master JEE Quadratic Roots. We highlight the main concepts, provide a list of examples with solutions, and include problems for you to try. Once you are confident, you can take the quiz to establish your mastery.
A root of the equation is a number (real or complex), say , which satisfies the equation i.e. . The roots of the quadratic equation with are given by .
JEE Conceptual Theory
As per JEE syllabus, the main concepts under Quadratic Roots are nature of roots, common roots, Vieta's theorem and symmetric function of roots, Newton's theorem, and location of roots.
Nature of roots
- : real and distinct roots
- : real and equal roots
- : non-real roots
- Conditions for rational, integral, and irrational, roots
Common roots
- Finding the conditions that the equations and have at least one common root or both the roots common.
- At least one common root:
- Both common roots:
Vieta's theorem and symmetric function of roots
- Vieta's theorem: If are the roots of the quadratic equation , then and
- Relation between roots and coefficients
- Symmetric function of roots:
Newton's theorem
- If and are the roots of and , then for any natural number we have
Location of roots
- Both roots greater than a specific number
- Both roots smaller than a specific number
- A specific number lies between both the roots
- Both the roots of are confined between two specific numbers and
- Exactly one root of lies between two specific numbers and
JEE Mains Problems
Find the value of such that the equations and have exactly one common root.
Concepts tested: Common roots
Answer: A) No such value of exists
Solution:
Using the condition of common root we have
But on putting in the given equations and we get the two equations as and respectively. Here we see that these two equations have become identical, i.e. both these equations will have both the roots in common.
Hence is neglected. And hence there is no such value of for which the equations and have exactly one common root.
Common mistakes:
- If you didn't check whether is associated with exactly one common root or both common roots, then you will end up saying that the given equations have exactly one common root for .
Find all the values of for which both the roots of the equation lie in the interval .
Concepts tested: Location of roots
Answer: B)
Solution:
Common mistakes:
If and are the roots of the equation , then find the value of .
Concepts tested: Vieta's theorem
Answer: A)
Solution:
Common mistakes:
The following is a problem from JEE-Mains 2015:
The following is a problem from JEE-Mains 2014:
JEE Advanced Problems
Find all possible parameters for which is non-positive for every .
Concepts tested: Location of roots
Answer: A)
Solution:
Common mistakes:
The following is a problem from JEE-Advanced 2015:
Let be the set of all non-zero real numbers such that the quadratic equation has two distinct real roots and satisfying the inequality . Which of the following intervals is/are a subset of
Note:
- Submit your answer as the increasing order of the serial numbers of all the correct options.
- For example, if your answer is then submit 12 as the correct answer; if your answer is then submit 234 as the correct answer.
The following is a problem from JEE-Advanced 2014:
The following is a problem from JEE-Advanced 2014:
Once you are confident of Quadratic Expressions, move on to JEE Quadratic Roots or JEE Complex Numbers.