SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE 

"Solving quadratic equations by completing the square" is one of the methods being applied to solve any kind of quadratic equation. 

First let us understand the method and look at some example problems for practice. 

Solving Quadratic Equations by Completing the Square Method

(i) First we have to check whether the coefficient of x² is 1 or not. If yes we can follow the second step. Otherwise we have to divide the entire equation by the coefficient of x².

(ii) Bring the constant term which we find on the left side to the right side.

(iii) We have to add the square of half of the coefficient of "x" on both sides.

(iv) Now the three terms on the left side will be in the form of a² + 2 a b + b² (or) a² - 2 ab + b².

(v) Then, we can write (a + b)² for a² + 2 a b + b² and             (a- b)² for a² - 2 a b + b². Then we have to solve for x by simplification.

Example problems on "Solving Quadratic equations by completing the square method"

(1) Solve the following quadratic equations by completing the square.

Problem 1:

Solve the quadratic equation  x² + 6 x - 7 = 0  by completing the square method 

Solution:

   (x + 3)² = 16

    x + 3 = √ 16

   x + 3 = ± 4

        x + 3 = 4             x + 3 = - 4

     x = 4 - 3                   x = - 4 - 3

       x = 1                       x = - 7

Problem 2:

Solve the quadratic equation  x² + 3 x + 1 = 0   by completing the square method 

Solution:

       x² + 3 x + 1 = 0

       x² + 3 x = -1

   x²+ 2 x x x (3/2) + (3/2)² = - 1 + (3/2)²  

   [x + (3/2)]² = - -1 + (9/4) 

   (x + (3/2))² = (9/4) - 1

   (x + (3/2))² = (5/4)

 (x + (3/2)) = √(5/4)

  x + (3/2) = ± (√5/2)

  x + (3/2) = (√5/2)                x + (3/2) = -(√5/2)

 x = (√5/2) - (3/2)                   x = -(√5/2) - (3/2)

   x = (√5 - 3)/2                          x = (-√5 - 3)/2

Problem 3:

Solve the quadratic equation  2 x² + 5 x - 3  = 0 by completing the square method 

Solution:

            2 x² + 5 x - 3  = 0

divide the whole equation by 2

        x² + (5/2) x - (3/2) = 0

        x² + (5/2) x = (3/2) 

        x² + 2 (5/2) x = (3/2)        

        x² + 2 x (5/2) + (5/2)² =  (5/2)²+ (3/2)

        (x + (5/2))² - (25/4)- (3/2) = 0

        (x + (5/2))² =  (25/4) + (3/2)

        (x + (5/2))² =  (25 + 6)/4

        (x + (5/2))² =  31/4

          x + (5/2) = √(31/4)

          x + (5/2) = ± √31/2

          x + (5/2) =  √31/2          x + (5/2) = -√31/2

          x = (√31/2) - (5/2)          x = (-√31/2) - (5/2)

           x = (√31 - 5)/2                    x = (- √31-5)/2

Problem 4:

Solve the quadratic equation  4 x² + 4 b x - (a² - b²) = 0  by completing the square method 

Solution:

4 x² + 4 b x - (a² - b²) = 0

dividing the whole equation by 4,we get

       x² +  b x - (a² - b²)/4= 0

x = (a - b)/2        or  x = (-a -b)/2

Problem 5:

Solve the quadratic equation    x² - ( √3 + 1) x + √3 = 0  by completing the square method 

Solution:

  x² - ( √3 + 1) x + √3 = 0

x = -1 , -√3

We hope that the students would have understood the stuff and example problems explained on "solving quadratic equations by completing the square".

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