Let's do an example first and then summerize what are the techniques to complete the square for quadratic equation in order to solve for x.

-2x^2 + 5x + 12 = 0

-2x^2 + 5x =  -12

-2 (x^2 - 5/2 x) = -12

so 2ab = 5/2 x  ("2ab" refers to the formula (a+b)^2 = a^2 + 2ab + b^2  or (a+b)^2 = a^2 - 2ab + b^2 )

Since a=1 (because x^2 is the a^2 in the formula), we get 2*1*b = 2b = 5/2 x, so b = 5/4

That means to complete the square we will add "b^2" (which is (5/4)^2 here) and then subtract (5/4)^2 in the parenthesis:

-2 [ x^2 - 5/2 x + (5/4)^2 - (5/4)^2 ] =  -12

Now that is same as  -2 (x - 5/4 )^2 + (-2) * (-5/4)^2 = -12

To simplify it we follow these steps:

-2 (x - 5/4 )^2 + 2 * 25/16  = -12

-2 (x - 5/4 )^2 + 25/8  = -12

Multiply both sides of the equation by 8 we get this:

-16 (x-5/4)^2 +25 = -96

then -16 (x-5/4)^2  = -121

 

Techniques summerization:

To complete the square for quadratic equation in order to solve for x, first we need to move the terms without x (which is "c" in the "ax^2+bx+c=0" form) to the right side of the equation. Then we divide both sides of the equation by "a" in the "ax^2+bx+c=0" equation. Through the formula (a+b)^2 = a^2 + 2ab + b^2  or (a+b)^2 = a^2 - 2ab + b^2 we will then figure out the value of "b", so then we know what is the b^2 we need to add then subtract. Finally just simplify the equation.

Hope this helps.