How do we solve quadratic equations by using the quadratic formula ?

General form of a quadratic equation : AX²+BX+C=0

Formula used to solve ANY quadratic equation:

X = -B + ­­­­Square root of ( B²-4AC)

       ———————————

                      2A

Step 1: Substitute

Step 2: SOLVE!

Step 3: Add

Step 4 : Subtract

Example :

X²-6X+5=0

X= -(-6)+SR( (-6) ²-4(1)(5)

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

              2 (1)

X= 6+SR (16)           

    -------------    =   

            2

6+4                                 

------  -->  5           

   2

  6-4

------   --> 1

   2

X : 5 , 1

Finally !!! 

How do we solve quadratic equations by taking square roots ?

Extra Information:

If B²-4AC = perfect square then two roots are rational.

If B²-4AC =not perfect square then it will become rational.

For Example :

Y= X²+6X+8

Substitute for B²-4AC : 6²- 4(1)(8) = 4 --> Rational

Solving Quadratic Equations :

Example :   X²+6=15

                  -15  -15

                 -----------------

               X²-9=0 -->  ( X-3) (X+3)  --> X = 3,-3

What To Do ?

Step 1 : Subtract so it becomes a zero .

Step 2 : Factor my equation that I got after subtracting

Step 3 : Do the Zero Product Property

Step 4 : Check if you want to be sure !

 

***FABULOUS! ***

                 

How do we solve quadratic equations by factoring ?

Zero Product Property states that when the product of 2 factors equal zero , than at least one of the 2 factors must equal zero.  EX: if 4 (  X-7 )=0 than 4 or ( X-7) must be zero . 

Step 1 : Factor if not already 

Step 2 :  Take each part of the factored equation and create an equation that equals to zero 

Step 3 : Solve ! 

Step 4 : Check by substitution if you are not sure .

EX : (X-1)(X-4)  ------ >    X-1 =0  and X-4=0 
                              +1  +1      +4 +4
                                              
                           X =1          X = 4 

Short Cut : When you reach the factored equation just switch the signs . You can do it mentally .

EX : ( X-6 ) ( X+2)  ------ > X=6    X=-2

How do we graph prabolas ?

A parabola is a U shaped graph . To graph a parabola you need to know how to identify the axis of symmetry and the vertex.

The Axis of Symmetry: the graph looks identical on both sides of the line of symmetry ( the axis of symmetry line )

Vertex : the Lowest point of the parabola that opens up , or the highest point of the parabola that opens down.  

                      Step 1 : Make a table

                    Step 2 : Graph the table 

This is what a parabola should look like , it also can be upside down .

         That is how you graph a parabola, Done ! Yay ! 

How do we solve rational proportions ?

Rational Proportions:

Technique: Cross Multiply

EX :

2N            8                  2N = 96

         =                 =                      

12             1                       2    

                       N = 48 

 

 

   

How do we add and subtract rational expressions ?

Adding and Subtracting Rational Expressions: @>------

*** Make sure you have common denominators ***

Steps :

1. Combine the numerators together
2. Put the sum found in Step 1 over the common denominator.
3. Reduce if necessary .




 



 Ex :   

Part 1 :   Adding Rational Expressions

10S               3S                  13

------     +                =               

 A - 2           A -2                A-2

Part 2 : Subtracting Rational Expressions

 12               10              2

------     -               =         

 G+8       G+8      G+8

Click the Picture ^ _ ^

How do we multiply rational expressions ?

Multiplying rational expressions is similar to dividing polynomials. All you have to do is factor and eliminate the terms. Here is an example to help you out :

X²+8x+15             (X+5) (X+3)        ( X+3)

                   • =                      =                                                                   

X²+10x+25            (X+5) (X+5)         ( X+5) 

If you are having any trouble look at the picture , it might help you out :