How Do We Solve Linear Quadratic Systems?

y = x^2 - 2x  + 2
y - 2x  =  - 2                                 Step 1:  First you have to change this into standard form : Y = MX +B )
  +2x          +2x
 ----------------
 Y = 2x - 2
      

2x - 2 = x^2 - 2x +2                                         Step 2)  Substitute this variable into the quadratic equation
-2x +2           -2x+2
---------------------
      0 = X^2 -4x +4                                                       Step 3)  Solve the quadratic equation

0=(x -   2)(x -   2)

   You have to make the equation equal Zero so if it`s positive to make it zero you have to make it negative, so if it`s negative you have to make it Positive.
For example :
                      0 = (x - 2)
                         X has to be positive 2

But if it was 0 = (x +2)
                        X would have to be negative 2

So the value of X is 2 so what you do next is      Step 4) Substitute x value into linear equation to get y value
                  y =  2x- 2 = 2(2)- 2 =  2
                     Solution =  (2,2)

How Do We use the discriminant to Find the Number of Solutions To a Quadratic Equation?

The solutions to a quadratic Equation is when the Parabola touches the X- axis which are also called the "roots".

 The Discriminant is B^2 - 4AC

 If B^2 - 4ac > 0 you`ll get two solutions
 If B^2 - 4ac = 0 you`ll get one solution
 If B^2 - 4ac < 0 you`ll get no solution

For example :
                         X^2 + 6X + 9 = 0
A = 1                  B^2 - 4ac = ?
B = 6                   6^2 - 4(1)(9)
C = 9                    36 - 36 = 0

         (One solution) because its equal to Zero

Another example:
                              -16x^2 + 11x - 11 = 0
A = -16                     B^2 - 4ac =?
B = 11                      11^2 - 4(-16)(-11)
C = -11                    121-+ 64(-11)
                                   121 + -704
                                          -583
           (NO solution) because its less then Zero

   Example:
                      5x^2 + 2x - 10 = 0
A=5                b^2 - 4ac
B=2                   2^2 - 4(5)(-10)
C=-10                 4- 200
                             204
           ( Two solutions) because its greater then Zero

How Do We Work with quadratics?

The first thing that you should think of when you think of quadratics is the 
      Quadratic Equation:
                 
So if you have the equation x^2 + 2x - 1

• First you have to find the value of A, B and C
• The value of A is in front of the first x 

            A = 1

• The value of B is in front of the second X                 

           B = 2
  The value of C is the number without an x after it
            C = -1
           Step 1: plug in      
        Step 2 : simplify                                                           

How Do We Simplify Radicals?

Step 1: Find all the " Prime Factors" starting with the smallest Prime 
Step 2: Rewrite using square Notations
Step 3: Separate square root notations 


Example: Simplify .

180 = 2·  90 = 2· 2· 45 = 2· 2· 9· 5 = 2· 2· 3· 3· 5

Therefore,

 = 2· 3 = 6.

here is a video to help you understand better

How do we Factor the difference of squares?

This expression is called a difference of two squares. 

The factors of          are and     

Example 1:

Factor:   x2 - 9Both x2 and 9 are perfect squares.  Since subtraction is occurring between these squares, this expression is the difference of two squares. What times itself will give x2 ?  The answer is x. What times itself will give 9 ?  The answer is 3. These answers could also be negative values, but positive values will make our work easier. The factors are (x + 3) and (x - 3). Answer:  (x + 3) (x - 3)  or   (x - 3) (x + 3)   (order is not important)

Here is a video to help you understand better....

How do we divide polynomials?

 Simplify 

Factor out the common factor from the top and bottom, and then cancel off:

X = 2




Here is a video to help explain how to divide polynomials:

How do we identify all the Elements of a Parabola?

When you have the formula to graph a Parabola you need to be able to understand what it stands for/ says....


                      y = ax² + bx + c
The  A for how wide or how narrow the parabola will be.
The   B stands for the line of symmetry
The  C tells you if its will be facing up or will be facing down.( If a Parabola is facing down the C is negative but if its facing up then the C is Positive)




To find the axis of symmetry you use the formula......






          X  = -  B
                       
                  2A


Roots are where and when the parabola touches the X-axis 

How do we Graph Parabolas?

Graphing the Parabola y = ax2 + bx + c


1.    Determine whether the parabola opens upward or downward.


                 a.    If a is positive, it opens upward.
                b.    If a negative, it opens downward.


2.    Determine the vertex.
             a.    The x-coordinate is .
                 b.    The y-coordinate is found by substituting the x-coordinate, from Step 2a, in the equation                                                     y = ax^2 + bx + c.


3.    Determine the y-intercept by setting x = 0.

by determining all the key features of the graph. That is, find the vertex, the x- and y-intercepts (if any), and additional points if needed. Find the domain and range of the function.

Opens upward since the coefficient of the x² is positive  

4.    Determine the x-intercepts (if any) by setting y = 0, i.e., solving the equation 
       ax2 + bx + c = 0.


5.    Determine two or three other points if there are no x-intercepts.



For Example:

Graph  

Vertex:  and  So the vertex is (3, -4).

y-intercept:  The y-intercept is (0,5).

x-intercepts:

                            

The x-intercepts are (5,0) and (1,0).

        Points plotted on grid below:

 Then connect the points with a smooth curve.
Here is A video to help you understand better how to graph a Parabola.

How do we solve rational Proportions?

Solve...
                   k+4             K+9
                --------  =    --------
                     2                 3
Step 1 : multiply each side by a number to get a common denominator

          3  *  k+4             K+9  *3

                --------  =    --------

          3*      2                 3      *3

Step 2:  you cancel the  denominator  because their the same 

          

                 3k+12          2K+18

                --------  =    --------

                     6                6

Step 3: start to subtract, add , divide, or multiply to both sides to cancel out terms to get x by its self and to equal to a number

          3k + 12 = 2k +18

        -2k           -2k

                                         

          k + 12 = 18

             -12     -12

                                         

                 k = 6

How do we solve rational Expression?

Simplify the following Expression.....

                                                2                  3

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

                                               A                  A^2

Step 1: Change fractions into equivalent fractions with a common denominator

                                      A *    2                  3

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

                                   A*      A                  A^2

Step 2: add or subtract  numerators, keep the common denominator

                                                2                  3                    2A -  3

                                            ------     -     -------       =   ------------

                                              A^2                 A^2                A^2

Step 3:Simplify the resulting expression, cancelling common factor

                               2A -  3

                           ------------     ( because the expression does not  simplify any more this is your final answer)

                                 A^2

How do we solve diamond problems?

First you will have a shape that look like a diamond (like the picture below). You will notice that 2 of the spaces are filled in. You have to use the pattern to be able to fill the other two out. The pattern is to have the two other numbers to add to the number on top and to be able to multiply to the number on the bottom. You do this by using the factors of the number you are going to multiply.
 
For example you will multiply 3 and 4 which gives you 12 which would go on the bottom. Then you add 3 and 4 which would be 7.
For example you have to figure out what can be multiply to get 5 and then what can also be added to 6. So first you use the factors of 5 which are 5 and 1 , so when you multiply 5 and 1 you get 5, and when you add 5 and 1 you get 6. So the only two numbers you can use to to fill the diamond out correctly if 5 and 1.

Classify Polynomials

How do we classify polynomials?

       First you find the degree of the polynomials for example: Fourth degree , cubic, linear quadratic, and constant. Then you have to figure out what kind of polynomial it is like a monomial , binomials ,or trinomial. You have to remember to put it in standard form which just means to put the polynomials in order from greatest to least by the exponents.