How can we solve any equation …

To apply completing the square method, the quadratic equation must be in the form of 

ax² + bx + c  =  0

Step 1 :

In the given quadratic equation ax² + bx + c = 0, divide the complete equation by a (coefficient of x²). 

If the coefficient of x² is 1 (a = 1), the above process is not required. 

Step 2 :

Move the number term (constant) to the right side of the equation.

Step 3 :

In the result of step 2, write the "x" term as a multiple of 2. 

Examples :

6x should be written as 2(3)(x).

5x should be written as 2(x)(5/2). 

Step 4 :

The result of step 3 will be in the form of 

x² + 2(x)y  =  k

Step 4 :

Now add y² to each side to complete the square on the left side of the equation.  

Then, 

x² + 2(x)y + y²  =  k + y²

Step 5 :

In the result of step 4, if we use the algebraic identity

(a + b)²  =  a² + 2ab + b²

on the left side of the equation, we get 

(x + y)²  =  k + y²

Step 6 :

Solve (x + y)²  =  k + y² for x by taking square root on both sides.