This page will teach you how to master JEE quadratic equations. 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.
An equation of the form with and is called a quadratic equation, where and are the coefficients of and and the constant term, respectively. The roots of the quadratic equation are given by .
As per JEE syllabus, the main concepts under quadratic equations are identity, discriminant, completing the square, factorization, repeated roots, and formation of new equations.
- is an identity if and have the same value for every real value. An equation with arbitrary coefficients may be an identity under certain conditions.
- real and distinct roots
- real and equal roots
- non-real roots
Completing the Square
- Completing the square:
Factorization and Repeated Roots
- is a factor of the polynomial is a root of the equation , i.e.
- and being roots of
- being repeated root of
Formation of New Equations
- Transformation of equation
For what real values of , will the equation have more than two solutions?
Concepts tested: Identity
For the quadratic equation to have more than two solutions, it must be an identity. Therefore, Solving these, we get
Taking the intersection of (1), (2), and (3), we get
- If you take the union of the values of which you got after equating the coefficients to zero, you will end up with option D), which is wrong.
Let be the roots of the equation , then find the quadratic equation whose roots are and .
Concepts tested: Transformation of equation
Let , where can take values . Then . On substituting the value of in , we get
Writing this in terms of the variable , we get .
- If you tried to find the new equation by using the exact roots of the given equation, it will lead you to a lot of additional, complicated calculations.
If and , provided that , then which of the following is true about the roots of the equation
Concepts tested: Discriminant
Answer: D) either two or four real roots
implies either or Let be the discriminant of the equation and be the discriminant of the equation . If then which implies has two real and distinct roots; if then which implies has two real and distinct roots. Hence, in any case, the equation has either two or four real roots.
- If you expanded the product of and , you would be left with a -degree equation and we don't know how to easily solve a -degree equation.
Find the sum of all possible values of for which the expression can be expressed as the product of three factors, two of which are identical and monic.
Concepts tested: Factorization
Step 1: Figure out the appropriate factors
Let the identical and monic factor be and then the remaining factor will be . So, we can say that .
Step 2: Comparing the coefficients and finding the values of
On comparing the coefficients, we get two equations : and . On finding the value of from first equation i.e. and substituting in the second one, we get and thus the corresponding values of are .
Step 3: Summing all the values
Hence the sum of the possible values of is .
- If you didn't compare the coefficients carefully, you will obtain incorrect equations and hence incorrect values of .
Suppose the inequality holds for all real and , then find the sum of all possible integral values of .
Concepts tested: Discriminant
Step 1: Finding sign of the quadratic equation in terms of
Step 2: Finding the sign of quadratic inequality in terms of
Step 3: Finding and summing the values of
The integral values of lying in the interval are . Hence the sum of all suitable values of is .
- If you thought that the coefficient of is 1, while when considered as a function in , the coefficient is .