1/2/13; How do we add and subtract rational expressions?

How do we add and subtract rational expressions?

Adding and subtracting rational expressionals are much like cross multiplying or adding and subtracting fractions , for example ;

First find the common denominator : 1/x + x/2y
the common denominator would be :2xy

 Next, after finding the common denominators remember that whatever you multiply to the bottom you multiply to the top as well : 2y/2xy +x^2 /2xy

Now combine the expression : 2y +x^2 /2xy and that would be your final result

12/4/12 how do we rationalize a denominator?

How do we rationalize a denominator?

There are two different ways you can rationalize a denominator , first understand that you can never have a radical as a square root for example : 5 over radical 9

- in order to solve this you would have to simplify the bottom by finding the square root so this becomes 5/3

Then you might come to more complex problems like 5/3+(rad.)2
the way you would solve this problem is by using the complex conjugate of the denominator so you would have (3+(rad)2)(3-(rad)2) then FOIL this
the foiled product would be ; 9-3(rad)2+3(rad)2-(rad)4 , next you need to combine like terms and the final product 11
Now you need to multiple the same number on the bottom to the top so it would be (5)(3-(rad.)2)
 this results to 15-5(rad)2/11 as the final product

11/28/12 how do we factor by grouping ?

How do we factor by grouping ?

- When you factor your taking out common terms so you need to group them , for example :
x^3+3x^2-4x-12
First factor the first half of the equation : x^3+3x^2
this becomes : x^2(x+3) because you take out the common factor and make it so that if you foil it you get what the original half of the equation was.
Second ,now you do the same thing to the other half of the equation: 4x-12
this becomes 4(x+3) because four can come out of each term
Third , you can now combine like terms (x^2 +4) (x+3)
but one of the terms can be simplified still so;
 The final equation becomes (x+2)(x+2)(x+3)

absolute value inequalities 9/13/12

How do we solve absolute value inequalities ?

-The purpose of absolute value is to change negative numbers to postive numbers
-When applying absolute value to inequalites you will have 2 'x' values representing the absolute value Example:
solve for 'x"1st:  l 5x-7 l>8
add 7 to both sides : l 5x l >15
then divide 5 from both sides to isolate the x :
l x l>3
Now you need to flip the signs:
 l 5x-7 l<-8
add 7 to both sides:
l 5x l <-1
now divide 5 from both sides to isolate 'x'
l x l <-1
Your results are :
l x l <-1
l x l <-8

imaginary numbers

How do we use imaginary numbers ?

-We define the square root of -1 as "i". This is also called an imaginary unit.
-We call the 'i' solution to the equation ; i^2+1=0

Simplfying imaginary numbers example
What is the square root of -25 ?
- first notice that there shouldnt be any negative numbers under a radical , this means that there is no real solution , so your going to solve for the no solution.
 so this will simplify to |25 and |-1
Now that 'i' is equivalent to the square root of -1 you can plug it in so the result will be 5i

What is the square root of -17?
this simplifies to |17 & |-1
now use 'i' in place of negative -1 and this becomes ; i|17

Notice that when the number is rational 'i' goes after the number and when the number is irrational the'i' goes before the number.




Note : there was no radical sign so I used "|" in place of it

The discriminant (10/23/12)

How do we apply our quadratic knowledge to real world situations? 

The quadratic equation :

Underneath the radical of the quadratic equation above, lies The Discriminant 
B^2- 4ac
Rules of the discriminant :
b^2-4ac>0 - you'll get 2 solutions
b^2-4ac=0 - you'll get 1 solution
b^2-4ac<0 - you get 0 solutions

For example:  x ^2 + 6x +9=0
label the letters to solve using the discriminant:
a=1
b=6
c=9
now plug it into the equation:
6^2-4(1)(-9) = 72
- referring back to the rules of the discriminant , the solution is greater than 0 so it has two solutions

-If b^2-4ac is a perfect square the two roots will be rational when you graph them
For example: x^2 +6x+8
a=1
b=6
c=8
plug into the formula : 6^2-4(1)(8)
the result is 4, since its a perfect square its rational

Next example: x^2 +3x-1
a=1
b=3
c=-1
plug into the formula :3^2-4(1)(-1)
this becomes: 9-4(1)(-1)
the result is 13 , since its not a perfect square its irrational


















Cited works :
Let's be clear,http://paulpietrzak.blogspot.com/2011/01/solving-quadratic-equations-quadratic.html

inverse of functions (10/3/12)

How do we calculate the inverse of functions?

- Two operations are said to be the inverse of each other if one of the operations undoes the other 

-The inverse of f(x) can be calculated by swtiching the x & y values
For example : What is the inverse if f(x)=2x?
 y=2x -------> x=2y
Then solve :
x/2=2y/2 ( the 2's on the right cancel out)
this then equals : x/2=y

- Using the inverse can also be applied to coordinates by switching the x & y as well
 For example : What is the inverse of { (3,2)(-2,5)(6,0)}?
This equals : {(2,3)(5,-2)(0,6)}