# COMPLETE THE SQUARE

"Complete the square" is one of the methods being applied to solve any kind of quadratic equation.

## How to do complete the square step by step

Step 1 :

First we have to check whether the coefficient of x² is 1 or not. If yes, we can proceed the second step, otherwise we have to divide the entire equation by the coefficient of x².

Step 2 :

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

Step 3 :

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

Step 4 :

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

Step 5 :

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.

## Complete the square - Examples

Example 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  =  1 x + 3 = - 4  x  =  -7

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

Example 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 = (√31/2) - (5/2)  x = (√31 - 5)/2 x + (5/2) = -√31/2 x = (-√31/2) - (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 "Complete the square".

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