CBSE Tutorials

Solution of a Quadratic Equation by Completing the Square

 

    Following steps are involved in solving a quadratic equation by quadratic formula

        ·    Consider the equation , where a ¹ 0

        ·    Dividing throughout by ‘a’, we get

            

        ·    Add and subtract , we get

            ,

            

        ·    If taking square root of both sides, we obtain

            

            Therefore

 

The Quadratic Formula: Quadratic equation, where a, b, c are real number and , has the roots as

            

Problems based on Solution of a Quadratic Equation by Completing the Square with Solutions

 

Problem:     

 

Solve by completing the square :

 

Solution:     

 

(i)    

    Þ    

    Adding i.e. on both sides, we have

    Þ    

    or    

    Þ    

    Þ    

    So, the solutions are         

 

Problem:

 

By using the method of completing the square, show that the equation has no real roots.

 

Solution:

 

        

    Þ                [Dividing the equation by 2]    

    Þ    

    Adding i.e. on both sides, we have

    Þ    

    Þ    

            

    Þ    

    RHS is negative but cannot be negative for any real values of x.

                    [ square of any real number is non-negative]

    Hence, the given equation has no real roots.

 

Problem:

 

Find the roots of the following equation

        

    by the method of completing the square.

 

Solution:     

 

We have,    

        

    Þ     [Dividing the equation by 4]

    Þ    

        Adding on both sides we have

    Þ    

                

    Þ    

    Þ            [taking square root of both sides]

    Þ    

    Þ    

    Hence, the roots are and

 

Problem:

 

Solve the equation by the method of completing the square.

 

Solution:     

 

We have,    

        

    Þ    

        Adding on both sides we have

    Þ    

    Þ    

                

                

    Þ    

    Þ    

    Þ    

    or     

        

    Þ    

    Hence, the roots are and 1.

 

Problem:

 

Using quadratic formula, solve the following quadratic equation for x:

    

 

Solution:     

 

Comparing the given equation

         with the standard quadratic equation

        

    We have        a = 1

                b = –2a

    And            

    Now Discriminant    

        

    Þ        Þ    

    Using the quadratic formula, we get

        

    Hence     or

 

Problem:

 

Using quadratic formula solve the following quadratic equations:

    (i)     (ii)

 

Solution:     

 

(i)    We have,    

            

        Comparing the equation with we have

             and

    \

    

                    [ square of any real number is non-negative]

    So, the given equation has real roots given by

        

    and,    

    Hence the roots are and -1

    (ii)    We have,

            

        Comparing this equation with we have

         and

    \    

    Þ    

    Þ    

    Þ    

    Þ        

    Þ        [ square of any real number is non-negative]

    So, the roots of the given equation are real and are given by

        

    and,     

        \     Solution is and

 

Problem:

 

Using the quadratic formula, solve the equation

    

 

Solution:     

 

    Comparing the given equation with

    We have

    Discriminant    

            

        

        

        

            [ square of any real number is non-negative]

    Hence, the given equation has real roots given by

                

    The roots are

         and    

    or             and    

    or                 and

        Hence, are the required solutions.