On wednesday, everyone was hastily finishing their radical equation assessments, so I will just be blogging for thursday. In class on thursday we learned how to solve Quadratic Functions by Completing the Square.
Completing the square gets its name because you can use a square as a visual to break down quadratics.
For example: x² + 10x + 25
http://www.helpalgebra.com/articles/completingthesquare.htm
An example of a perfect square equation is (x-1)² = 25 where both sides of the equation are perfect squares so you can square root both sides of the equation.
√(x-1)² = √25 x-1= +/- 5
x= 6 and x= -4
☞note: if you start with a square, you should not have any extraneous solutions.
But if you have an equation like x² -10x - 18 = 0, where both sides of the equation are NOT square roots, and the equation is not factorable, you must follow these steps.
Here is an example problem:
x² - 10x - 18 = 0
Can this equation be factored? You want to make this equation factorable in ax² + 2ab + b² form.
a = 1
b = -10
c = 18
First, the third term (18) must be brought over to the other side. The mystery number will make the first side factorable. You add it to the other side as well because then you are not actually changing the equation, because the numbers would cancel. As a rule in algebra, what you do to one side, you must do to the other.
x -10x + ? = 18 + ?
To find your mystery number, divide b, which in this case is -10, by 2, and then square it. (b/2)² will always give you your mystery number.
(-10/2)² = 25
Now your equation is factorable.
x - 10x + 25 = 18 + 25
(x-5)² = 43
Square root both sides of the equation.
√(x-5)² = √43
x-5 = +/- √43 ☞note: all square roots are positive and negative
43 is not a square, and is not divisible by any squares, so you leave it inside the square root prison.
Finally, Isolate the x.
x = 5 +/- √43
Here is another example of how to do an equation that is not a perfect square equation:
x² +4x -96 =0
Bring the 96 over to the other side.
x² + 4x + ? = 96 + ?
(4/2)² = 4
x² + 4x + 4 = 96 + 4 (which conveniently adds to 100 which is a square! Its always nice when math works out in whole numbers☺)
Square root both sides.
√(x+2)² = √100
x+2 = +/- 10 ☞note: both positive and negative 10 are square roots of 100
x = +/- 10 - 2
x= 8 and x= -12
What do you do if the x² has a coefficient?
3x² + 5x -7 = 0
3x² + 5x -7 = 0
If the x² in your trinomial square has a coefficient, you have to divide the equation by the number of the nasty coefficient to cancel it out.
3x² + 5x- 7 = 0
_____________
3
x² + 5/3x - 7/3 = 0 ☞note: 0 divided by anything is 0 (remember 0/K and N/0? 0 divided by anything is 0, but it is impossible to divide anything by 0)
x² + 5/3x + ? = 7/3 + ?
(5/2 • 1/2)² = (5/6)² = 25/36
x² + 5/3x + 25/36 = 7/3 + 25/36
(x + 5/6)² = +/- √109/36
x = (+/-√109/ 6) - 5/6
x = +/- √109 - 5
__________
6
The Quadratic Formula and Completing the Square
Does everyone remember the quadratic formula? Well, here's how the quadratic formula ties into completing the square.
ax² + bx + c = 0
Divide the equation by a to get the x² on its own.
ax² + bx + c = 0
_____________
a
x² + (b/a)x + b²/4a² = -(c/a)x + b²/4a²
(x + b/2a)² = -4ac/4ac² + b²/4a²
Square root both sides.
x + b/2a = √(b² - 4ac)/4ac looking familiar?
x + b/2a = +/- √b² - 4ac
____________
2a
x = -b +/- √b² - 4ac
_____________
2a
yay!!
Here is a video about the Quadratic Formula. You may remember it if you were in Dubuque's math class last year. Beware the end...
x² + 5/3x + ? = 7/3 + ?
(5/2 • 1/2)² = (5/6)² = 25/36
x² + 5/3x + 25/36 = 7/3 + 25/36
(x + 5/6)² = +/- √109/36
x = (+/-√109/ 6) - 5/6
x = +/- √109 - 5
__________
6
The Quadratic Formula and Completing the Square
Does everyone remember the quadratic formula? Well, here's how the quadratic formula ties into completing the square.
ax² + bx + c = 0
Divide the equation by a to get the x² on its own.
ax² + bx + c = 0
_____________
a
x² + (b/a)x + b²/4a² = -(c/a)x + b²/4a²
(x + b/2a)² = -4ac/4ac² + b²/4a²
Square root both sides.
x + b/2a = √(b² - 4ac)/4ac looking familiar?
x + b/2a = +/- √b² - 4ac
____________
2a
x = -b +/- √b² - 4ac
_____________
2a
yay!!
Here is a video about the Quadratic Formula. You may remember it if you were in Dubuque's math class last year. Beware the end...
Here is a khan academy video on completing the square. I often use his videos for review, and find them really helpful.
Problems using these concepts are on page 294 in the book.
Hope my blogging was satisfactory!
-Grace
This was a really good blog! The visual was very good in the beginning, and the explanation was very helpful. The example problem and explanations to help understand the steps helped clear up any confusion. It was also very helpful to give examples of different (more difficult) solutions along with the simpler ones. Overall a very thorough blog!
ReplyDeleteGreat job Grace. The picture of completing the square was nice because it allows everyone to really see the roots of how to solve the equations. As you went through example problems, you added in tips and rules right near the problems, making it easy to see which rules connected to each problem. Your organization and structure of the blog was great as well. Good job!
ReplyDeleteNice blog! It was very informative and will be helpful in studying for our test later this week. I liked how you tied in the method of completing the square with the quadratic formula. The illustration for completing the square was also a nice refresher. On some of the more complicated problems might have been helpful to have them written out by hand so they are easier to follow but other then that this was a great blog.
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ReplyDeleteThis was a great blog it contained all the information we have learned and it'll be a good source to my studying for the test tonight. i like how you had the visual for completing the square which helped explain exactly how the formula works. On some of the more complex problems it would have been great if it had been written by hand instead of typed.
ReplyDeleteI really liked this blog! I really liked that you inserted the videos and gave some information about it before we saw it so we could keep track and understand the overlaps. The visuals were really goo throughout the whole post. It was very thorough.
ReplyDeleteThis was a very good blog and informational! It really helped with some of the stuff that we recently did and it helped with tying some of the things that we have learned together! The videos really helped and it was nice that you took the time to find all of them! Also the visual square at the beginning really helped with many of the concepts! great job.
ReplyDeleteThis blog was really helpful! It's very detailed and neat and goes over everything really well. This is going to be good to use to study for exams. The square at the beginning was great to visualize what we talked about with completing the square. The only little thing I would say is that it was a little difficult to follow the typed out problems. Writing them out by hand or on notability or something would have made it easier to follow, but the problems are really helpful examples. I like how you used multiple methods of presenting the topic (the picture, example problems, and videos). Awesome blog!
ReplyDelete