A perfect square has the form .

In this case, , so  is  or .

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

To solve the equation by completing the square first move the constant term to the right hand side of the equation.

Now, remember to divide the middle term by two. Then square it and add it to both sides of the equation.

From here write the the middle term divided by two in a binomial expression

Square root both sides and recall that .

Solve by completing the square

add to both sides, where .

Factor

Add 7 to both sides:

Divide both sides by the coefficient on x^2:

Add  to both sides:

Form the perfect square on the left side:

Simplify the right side:

Take the square root of both sides:

Solve for x:

To complete the square, we must remember that our goal is to make a perfect square trinomial out of the terms we have.

We are given a function that we must set equal to zero if we want to find its roots:

Now, subratct 1 to the other side so we have only x-terms on one side:

Now, on the left side of the equation, in order to make a perfect square trinomial, we must take the coefficient of x - in this case, -6, and divide it by two, and then square that number:

This term becomes our "c" for the trinomial . However, because we introduced this new term on the left side of the equation, we must add it to the right hand side as well, so that we aren't "changing" the original equation:

Next, we can convert the perfect square trinomial into the square of a binomial:

This comes from the definition of the binomial, squared. When we FOIL (or use the memory tool "square the first term, square the last term, multiply the two terms and double") we get our original trinomial.  

Now, to solve for x, take the square root of both sides, and add three to the other side:

.

To complete the square for any expression in the form , you must add .

In this case, 

To complete the square for any expression in the form , you must add .

In this case, 

Vertex form for a parabola is

where (h, k) is the vertex. 

We start by eliminating the leading coefficient by dividing both sides by 3.

We now subtract 6 from both sides to set up our "completing the square" technique.

To complete the square, we divide the x coefficient by 2, square the result, and add that result to both sides.

Since the right side is now a perfect square, we can rewrite it as a squared binomial.

Solve for y by adding 2 to both sides, then multiplying both sides by 3.

First subtract over the constant term just to get the x terms by themselves.

Now use the property of completing the square. In completing the square we take the constant from the "x" term (Not the x² term). We take the constant, in our case 3, then we divide it by two. This is going to be the constant in the perfect square!

Now, square this constant and add it to both sides:

This is our new constant that we can use to complete the square. Add this constant to both sides to get an equation that looks like this:

Because of the completing the square method, we know that our constant within the square should be 3/2 just like we found before, so now we can write down the completed square, and we are done.

Simplify the right hand side:

First, get the x² term by itself. Do this by dividing every term by 4,

Now subtract over the constant term just to get the x terms by themselves.

Now use the property of completing the square. In completing the square we use the constant from the "x" term (Not the x² term). Take the constant, in our case 4, then we divide it by two. This is going to be the constant in the perfect square!

Now, square this constant and add it to both sides:

This is our new constant that we can use to complete the square. Add this constant to both sides to get an equation that looks like this:

Because of the completing the square method, we know that our constant within the square should be 2 just like we found before, so now we can write down the completed square, and we are done.