In this video, we are going to look at how to solve equations by completing the square.
If we are given
we can try to factor it, but it would not work. This is when we would solve by completing the square. First, we want to get rid of the last term, so we will add 11 to both sides, giving us
Now we can complete the square.
Since we want to add 4 to the left side to complete the square, we must also add 4 to the right side to keep the equation the same. At this point we have
Now if we factor it, we have
Or it can also be written as
From here we take the square root of both sides
Solve for x by subtracting 2 and the final answer is
Let’s go over solving equations by completing the square.
Just to recap, if we have this:
We can solve by getting their square roots.
To solve for ,
So we have two answers:
Our solution set is
But sometimes we are not given equations in this form.
But we could translate or rewrite quadratic equations into this form and then use this method.
The way to do that is to complete the square.
We have this quadratic equation:
Let’s try to factor this.
But there are no two numbers that will add up to and when multiplied the answer is .
So we cannot factor this one.
That’s why we’re going to solve it using completing the square.
We have to look at
to determine what number needs to be added here to complete the square.
The first thing we need to do is to get rid of .
Now let’s go back to the formula since we have -term
So we’re going to add on both sides of our equation.
Now, we can factor this.
Now let’s solve for :
So our solution set is
It’s possible that when we solve the square root we’ll have an answer like this. Where there’s an integer plus or minus a radical term.
If these came out as integers and not this type of terms, that means we could have factored it in the beginning.
Let’s have an example like that.
Let’s factor this:
Let’s solve it
Our solution set is
This is our previous lesson.
Let’s solve this by completing the square.
Let’s now use the formula
Let’s go back to solving and add in the equation
Let’s solve the terms on the right side.
Now, let’s factor the equation
Then let’s find its square root
Then let’s solve for :
So we have two solutions
Our answer is the same when we did factoring earlier.
In this case, we could agree that it’s easier to factor than to complete the square.
But of course, the rule for completing the square is true.