Completing the Square

Hi Everyone!
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

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...

           

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