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