Sunday, February 22, 2015

Completing the Square of a Quadratic Function in song


Completing the Square Jingle
You half it,square it add it to both sides.





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 by adding 16 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 .In order to undo a perfect square we will take the square root of both sides, and 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 ,and you have to account for both possibilities. Now we just solve for x ,and 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 4 plus 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.
Finally, 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.
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 .
Finally we just add our answer and simplify our radical and there are two solutions. That is how you complete the square using our jingle.


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