Rationalizing Imaginary Denominators

When dealing with fractions with imaginary denominators, one must express this as a complex number in a+bi form.


For example:

17 - 3i
10 + 7i

1) Multiply by the conjugate of the denominator over itself (fraction equals 

17 - 3i   *   (10 - 7i)         
10 + 7i       (10 - 7i)

2)Foil. FOIL the top and FOIL the bottom.

170 - 119i - 30i + 21i²
100 -70i + 70i + 49i²



3)Multiply top by top, bottom by bottom.



170 - 149i + 21i²                 
    100 + 49i²                                

                                                                               
4)Combine the imaginary terms. They (i) 

cancels out in the denominator.


170 - 149i - 21
    100 + 49

5) Combine constant terms on top and bottom.


149 -149i
      149

6)Split apart into a+bi form.

149  -  149   i
149      149



1 - i

Finding the roots of a parabola.

To find the roots of a parabola is fairly easy. Its all about the formulas.

Questions can be asked like this.

Determine the roots of the equation x2−2x−3=0.

*The trick is to see whether an equation is factorable or not.

x^2 - 2x - 3 = 0    Factorable.

(x - 3) (x + 1)      BAM!

Now find the roots.

x - 3 = 0    x + 1 = 0

x = 3        x = -1

The roots of the equation are 3 and -1.

Now if you get an equation that isn`t factorable like this -x^2 + 4x -3 = 0

Use the quadratic formula.

     a = -1, b = 4, c = -3

-----------------------------------------

-4 + or - √ 4^2 - 4(-1)(-3)

_____________________

                                                                     2(-1)

                  

                                                           -4 + or - √ 16 -12

                                                                   _____________________      

                                                                                        -2

                                                            -4 + or - √ 4                       -4 + or - 2                               

                                                            _________                       __________

                                                                     -2                                       -2

-4 + 2                    or                   -4 - 2

 __________________________________________                                                                   

     -2                                                   -2

-2                                                         -6

__                                                      ____

-2                                                         -2

ROOTS are 1 and 3.  DONE!

Systems of equations :$

*Systems of equations is asked when there are two equations. One must use both to find a coordinate; x and y.
*This is called solving linear systems algebraically.


Solve the system of equations 2x - 3y = 23  +3y=23and -x- 6y= -4 
by combining the equations.

1) First eliminate x.

2x - 3y = 23   -x - 6y = -4

Use both 2 and -1, to eliminate x.

1(2x - 3y = 23)   2( -x - 6y = -4)    Distribute.

2x - 3y = 23

-2x - 12y = -8           2)Make sure to eliminate x, one is positive and one is negative.

____________

0x - 15y = 15

     _________

       -15    -15

y = -1                3) Now use y to find x. Plug in -1 to an easy equation.

2x - 3(-1) = 23

2x + 3 = 23

     -3      -3

__________               

2x = 20

______

    2

x = 10

(10,-1)

Done!

Simplifying Expressions with Rational Exponents =}

Rational exponents can be turned into radicals.

x^1/2 = √x
x^1/3 = ³√x
x^1/4 = ⁴√x

Examples:

1) 27^1/3 = ³√27 = ³√3^3 = 3

2) 12^1/2 * 3^1/2 = √12 * √3 = √36 = 6

3) 16^1/3 * 4^1/3 = ³√16  * ³√4 = ³√64 = 4                

Remember Simplifying:

√2² x² y² z²  =  2xyz

x³ x ^{4 =} x<sup>7      

More Examples: 

1) (x^4 y ^6 z^8) ^1/2
x^4/2 y^6/2 z^8/2</sup>
x² y³ z <sup>4

2) 4^3/2
(4^1/2)^3
(2)^3</sup>
 8

3) (4^3)^1/2

64^1/2
√64
8

Factoring by Grouping =!

How do we factor by grouping? 

In order to factor by grouping, basic simplifying and factoring is needed.

Simplifying: clearing parentheses, combining like terms by adding coefficients then combining constants. 

For example.

3(x+2) + 5(x+2)

First distribute to clear parentheses.

3x + 6 + 5x + 10

Next combine like terms by adding coefficients.

8x + 16

_____________________
Now do some Factoring.

x² - 5x -36

(x - 9)(x + 4)



 Back to Simplifying: Now try grouping instead of combining like terms.

3(x + 2) + 5(x +2)

Now ask: What does this equation have in common with itself?

The answer is : (x+2)

So place 3 and 5 in parentheses as well.   

(3+5)(x+2) *Don`t foil.

Then you`ll get 8(x+2)     
______________________

Now for a problem involving Factoring by Grouping.

x³ + 3x² - 4x - 12

• Separate the equation 

x³+ 3x² |  -4x -12

• Simplify each side

x² (x + 3) |  -4 (x +3)   

• Trick is to have the (x + 3) on both sides. Ask: What do they have in common?
• Then group.

(x²-4)(x+3)

• Factor (x²-4) to get the perfect answer.

(x+2)(x-2)(x+3)      

Functional Notation *---*

Function Notation is just using a symbol such as f(x) =, as another way to represent y.
Simply all that is needed is for evaluating each function is substitution.

For example:

(the arrow is read "is mapped to")

(the vertical bar is read "such that")

So using an equation f(x) = x² + 7


Find f(3).

To find f(3) replace x with 3.

f(3) = (3)² + 7

The answer is 16.
Courtesy of http://www.regentsprep.org/Regents/math/algtrig/ATP5/EvaluatingFunctions.htm


                         

Finding: Axis of Symmetry & Vertex

Axis of Symmetry: is a line that cuts through a figure to create two sides as a mirrored image.

That line is the axis of symmetry.

Finding the axis of symmetry is commonly used in quadratic equations, to create parabolas.

~Axis of symmetry is fairly easy as long as one knows this formula.


*For example: y=x^2 + 6x+8

(a = 1, b = 6, c = 8)

-b/2a = -6/2(1) = -3

x = -3


Axis of Symmetry is x = -3
 Yay!











Vertex is the turning point of a parabola. Mainly the point where the parabola created its U-shape.

The Formulas one needs for a vertex would be...

     &

For example y = -x²+4x+1

a = -1, b = 4 , c = 1                               axis of Symmetry is x = 2.

x = -b/2a                                       


x = -4/2(-1)                                                          

x = 2

Now one would substitute x for 2 in the quadratic equation y = -x²+4x+1.

y = -x²+4x+1
y = -(2)² + 4(2) + 1
y = -4 + 8 + 1                                      Vertex is (2,5)
y = 5

Now x and y were found. :P

Imaginary numbers? o.O

How do we use imaginary numbers?

Sometimes in Trigonometry, one will have a square root that is negative.

Some thing like this. 

*I know.

* Commonly used in Quadratic Formulas, Imaginary numbers are used to simplify a square root though it is (negative).

*By using an [i] you are able to identify its square or if irrational its simplified form.

*Remember the square root is considered a real number [-36], and [6i] is imaginary

6i

Now one can get something like this, which is irrational.  

         

√-18

√-1 • √18

(√18 = 9•2 = 3•3•2)

i • 3√2

[3i √2]   

 Courtesy of  http://www.regentsprep.org/Regents/math/algtrig/ATO6/SquareRootLes.htm

Flipping inequality symbols :D

"Why do we flip the inequality symbol when multiplying by a negative number or solving absolute value inequalities?"

When one has an absolute value inequality, one must switch the inequality symbol when multiplying by a negative number.

> First always remember that only when multiplying or dividing symbols are switched.
>The reason is because if one does not switch the inequality symbol (depending if its negative) it might not be true.
> Also to address a number line one must make sure a switched is performed.

lets try some and see
5 - 3x [>] 20
try -4
5 - 3(-4) [>] 20
5 + 12 [>] 20
17 [>] 20 

*as you can see not true since the symbol wasn't switched

*Here is how it does work.

5 - 3x [>] 20

 Try -4 ( and switch)

5 - 3(-4) [<] 20

5 +12 [<] 20

17 [<] 20

Courtesy of http://www.algebra.com/algebra/homework/Inequalities/Inequalities.faq.question.101178.html