Hello welcome to MooMooMath
Today we are going to talk about "Completing
the square"
I have a little song I'm going to share with
you to help you learn complete the square.
So when do we use complete the square?
We use it to solve a quadratic equation.that
may not factor.
One of the ways to solve it is to use the
quadratic formula, but this is just another
way you can solve a quadratic by using the
completing the square.
You will also use it later in Math with Conic
sections to change them to different forms.
Let's get started with learning "Completing
the Square"
Here is our quadratic x^2-8x-4=0 I will write
above it the standard form which is ax^2 +
bx + c = 0
Now we are going to start the process of learning
completing the square,and I have a little
song to help.
This is how the song goes.
You half it,you square it,and you add it to
both sides.
You half it,you square it,and you add it to
both sides.
Now I have taught this for a long time and
students tend to remember this little jingle
in their mind to remind them how to complete
the square.
What are you taking half of?
You are taking half of the middle term,the
b term,the coefficient, and you are dividing
it by two,and then you are taking that term
and squaring it,and then adding that to both
sides of the quadratic equation because it
is an equation and you have to keep it equal.
Let's run through an example problem.
x^2-8x-4=0
The middle term is 8 so we will half it, which
equals negative 4,and then square it,and that
gives us 16 and then we will add 16 to both
sides of the quadratic equation.
Now let's apply it.
x^2 -8x -4 =0 the 4 does not help us complete
the square so will move it to the other side
of the equation (kick it over the fence) to
the other side of the equation So we will
set it up like this, a square term,a linear,a
space and then our 4 We half it,we square
it,and we add it to both sides.
So now we will add to both sides.
Add16 to the left and 16 to the right side
because you have to keep it balanced.
Now what we have done with completing the
square is we created a perfect squared trinomial
on the left side, This trinomial will factor
down to a perfect square.
What multiplies to 16 but adds to negative
8 ? Negative 4 times negative 4 equals 16
and adds to negative 8.
So this will factor to x-4 times x-4,but we
don't want to write it this way.
Instead we will write it (x-4)squared.which
is a perfect square.
We are left with (x-4)squared = 20 Now we
will apply the square root.
method.
We will take the square root of both sides
in order to solve for x To undo a perfect
square we will take the square root of both
sides,We are left with x-4 and on the right
side of the equation we take the square root
of 20.
You have to think of the positive and negative
solutions You have to account for both possibilities.
Now we just solve for x so we will move the
4 to the right and simplify our answer.
You are left with
x=4 plus or minus square root 20.
This answer is not simplified because I can
bring down the square root of 20 into 4 times
5 which simplifies into 4plus or minus 2 square
root 5.The two solutions are 4+2 square root
5 and the other solution is 4 minus 2 square
root 5 Again,make sure you think of both solutions.
That is how you solve a quadratic using the
complete the square method.
Let's review the song.
We half it,we square it,and we add it to both
sides.
We half it,we square it,and we add it to both
sides.
then you finish up by taking the square root
of both sides,rewrite the left side as a perfect
square trinomial,then factor it in factor
form,take the square root of both sides with
a positive and negative,and then solve for
x.
Let's go over the steps one more time
We took the b term and halved it,squared,and
added it to both sides.
Then we factored the left side and created
a perfect square.
We then took the square root of both sides.You
are left with x-4 =plus or minus square root
of 20 and account for both solutions and get
the x by itself by adding the 4 and then we
just add our answer and simplify our radical
and there are two solutions.
That is how you complete the square using
our jingle.
