# How do you solve an equation by completing the square?

##### 2 Answers

(This is going to take a minute or two.)

**Completing incomplete squares:**

**Background:**

The square of an expression of the form

Notice: the sign on the middle term matches the sign in the middle of the binomial on the left AND the last term is positive in both.

Also notice that if we allow

**Completing the square:**

An expression like

Of course you can's just add a number to an expression without changing the value of the expression, so if we want to keep the same value we'll have to make up for adding

We do this by also subtracting

We write

And

**Solving an equation** by completing the square:

Solve:

Each of the following equations is equivalent (has exactly the same solutions) as the lines before it.

And the last equation above is satisfied exactly when:

The solution set to the first equation is:

I'll post another (more challenging) example too.

**Second Example**

(You should probably read the first one first.)

Solve by completing the square:

Do you see what we did there? We factored out a

Now we will complete the square **inside** the parentheses.

The middle term is

So we must have

Keeping the perfect square together, we re-write this as:

(Note the signs in the middle and the return of the

This will be true exactly when:

Or, better yet: