To complete the square, we need the left side in a form  or  so that we can factor it into form  or .

To do this, we first divide out three on the left-hand side to obtain:

We then take 1/2 of the number in our  term (in this case ) to obtain :

We then must add this to each side, but because we are completing the square inside of a parenthesis which is being multiplied by 3, we don't add 36 to each side, but rather 3 times 36, or 108. Therefore, we obtain:

 and 

Once the polynominal is factored out and everything is moved to the left, the equation becomes  which does not factor evenly, so you could use the quadratic formula, or complete the square. To complete the square, a coefficient must be found to factor the polynominal into a perfect square. The polynominal  factors to  so we know that  so  and . To complete the square you add and subtract  from the left side of the equation and strategically place parentheses to get , this simplifies to , which simplifies to , 

When using the complete the square method we will divide the  coefficient by two and then square it. This will become our  term which we will add to both sides.

In the form,

our ,  and we will complete the square to find the  value. 

Therefore we get:

In order to complete the square, we must get all the terms with x in them to one side of the equation. For this problem, we subtract 9 on both sides in order to yield:

.

Remember that completing the square means that we take half of the coefficient of x (), square this new value (), and add it to both sides of the equation. Our equation now looks like:

The left side is a perfect square polynomial, as we have set it up this way. We factor it as such: 

After some cleanup, we arrive at:

In order to solve for x, we must take the square root of both sides of the equation. 

Finally, we add 6 to both sides of the equation, and simplify the square root of 27:

To complete the square, first set our equation equal to 0:

add 7 to both sides

we want to find the number we can add to both sides so that the left side can be factored as one binomial squared. This binomial must be , since when multiplied by itself you'd end up adding which is what we need. Multiplying: 

so the number we want to add to both sides is 6.25

we constructed the left side so we could re-write it as:

simplifying the right gives

now we can subtract 13.25 from both sides to get 0 again:

so our equation is

To complete the square, first set the equation equal to 0:

subtract 1 from both sides

factor out the 2 on the left side

Now we're trying to figure out what we can add in that space so that the expression in parentheses can be factored as a binomial squared. works  as the binomial since we know we will be adding . Squaring this yields:

So the number we want to add is 1. BUT BE CAREFUL! We're adding a 1 to inside those parentheses, so really, we're adding 2, since we distribute that 2. Add 2 to both sides:

we can easily simplify the right side, plus we know that we can factor the left since we set it up to be able to:

 now subtract 1 from both sides

so our equation is

To solve a quadratic with an "a" term of 1 (from the standard form ) by completing the square you must first move over the constant. Next, halve the "b" term, square it, and add to both sides. Then factor the left side and set it equal to the constant. Note that the factor of the quadratic you "made" will always be of the format  where the sign is the original sign of the b term.

Move the constant:

Halve the b term, square it, and add to both sides:

Factor the left side and simplify the right:

Take the square root of both sides:

Solve for x:

To solve an equation by completing the square, you must factor the perfect square. The factored form of  is . Once the left side of the equation is factored, you may take the square root of both sides.

First, factor out the 2 from the first 2 terms:

add 3 to both sides

inside the parentheses, add

since the 4 was added in the parentheses, it's multiplied by 2. That means we added 8, so add 8 to the other side too

simplify by re-writing the left and adding 3 and 8 on the right

subtract 11 from both sides

A perfect square has the form .

In this case, , so  is  or .

To square a fraction, simply square the numerator and the denominator: .