CBSE Tutorials

Nature of Roots of a Quadratic Equation

    In previous section, we have studied that the roots of the equation are given by

    

    A quadratic equation has

·    Two distinct real roots if .

    If we get two distinct real roots and

·    Two equal roots, if .

    If then

            i.e.    

            So, the roots are both

·    No real roots, if

    If then there is no real number whose square is

 

Note: determines whether the quadratic equation has real roots or not, hence is called the discriminant of quadratic equation.

It is denoted by D.

 

Solved Problems based on Nature of Roots of a Quadratic Equation

 

Problem:    

 

Find the discriminant of the quadratic equation and hence find the nature of its roots.

 

Solution:     

 

The given equation is of the form where a = 2, b = – 4 and c = 3. Therefore, the discriminant.

        \    

        So, the given equation has no real roots.

 

Problem:    

 

Find the discriminant of the equation and hence find the nature of its roots. Find them, if they are real.

 

Solution:     

 

Here and

        Therefore, discriminant

        Hence, the given quadratic equation has two equal real roots.

        The roots are i.e., i.e.,

 

Problem:    

 

Find the values of k for which the following equation has equal roots:

        

 

Solution:     

 

We have,

            

        Here,     and c = 2

        \    

        Þ    

        Þ    

        The given equation will have equal roots, if

             Þ Þ k – 12 = 0 or k – 14 = 0

                Þ k = 12 or, k = 14

 

Problem:

 

Find the values of k for which the given equation has real roots:

        (i)         (ii)     (iii)

 

Solution:     

 

(i)    We have

                

            Here,     and c = –2

            \    

            The given equation will have real roots, if

                

        (ii)    The given equation is

            Here,    a = 9, b = 3k and c = 4

            \    

            The given equation will have real roots, if

                

            Þ    

            Þ    

            Þ                

            Þ            

        (iii)    The given equation is

            Here,     and c = 1

            Þ    

            The given equation will have real roots, if

                

            Þ    

            Þ            

 

Problem:

 

Find the values of k for which the equation has no real roots.

 

Solution:     

 

The given equation is

        Comparing the given equation with we have a = 1, b = 5k, c = 16

        

        The given equation will have no real roots if D < 0

        Þ    

        Þ    

        Þ             [If ab < 0 and a > 0, then b < 0]

        Þ            [If then –a < x < a]

 

Problem:    

 

If – 4 is a root of equation and the equation has equal roots, find the values of p and q.

 

Solution:     

 

Since – 4 is a root of we have

            Þ    

        Þ        Þ                    …(i)

        Putting p = 3 in equation we have

        

        Equation will have equal roots if D = 0 i.e.

        Þ        Þ             [Using (i)]

        Þ    9 – 4q = 0    Þ    

        Hence, p = 3 and

 

Problem:

 

If the roots of the equation are equal, prove that

 

Solution:     

 

The given equation is

        Comparing the given equation with

        We have, and

        

        For real and equal roots,     \

        Þ    

        Þ    

        Þ    

        Þ    

        Þ    

        or     

        or         

 

Problem:    

 

If – 5 is a root of the quadratic equation and the quadratic equation has equal roots, find the value of k.

 

Solution:     

 

Since – 5 is a root of the equation Therefore,

            

        Þ    

        Þ    

        Þ    

        Putting in we get

            

        This equation will have equal roots, if

            Discriminant = 0

        Þ    49 – 4 ´ 7 ´ k = 0

        Þ