Another blog

Aim: How do we calculate quadratic inequalities?

Hi :) again, another math blog.

what are quadratic inequalities?
quadratic inequalities are similar to quadratic equations the only difference is the inequality signs are added.

A sample of a quadratic inequality is :
x2+10x+25≤ 0

lets use this quadratic inequality to solve it :

STEP 1: Foil the quadratic inequality as a quadratic equation.

x2+10x+24 =  (x+6)(x+4)

STEP 2: Find  X by equaling the expressions to zero.

     x+6=0                                              x+4=0
        -6=-6                                               -4=-4
         x=-6                                                x=-4

STEP 3: Plot points back to the equation to know what way the arrow will be going.

Ex: (0,0)
0^2+10(0)+24 ≤ 0
0+0+24≤ 0
24 ≤ 0

This is not true because 24 is greater than 4, therefore the (-4) will not go in the direction that the zero is at.

Ex: (-10,0)
-10^2+10(-10)+24 ≤ 0
100+(-100)+24≤ 0
 24≤ 0

Because 24 is greater than 0 the closest number to -10 ,(-6) will not go in the direction of -10. 

With the new found information we will now use it and correctly plot the points in a graph.

STEP 4: GRAPH 
(Remember )
≤   Less than or equal to signs have a closed circle when graphed.

YOU TRY NOW :

1.

Using a number line,  graph the solution set of

2.

Solve algebraically:

Works Cited:

Regents prep Problems: http://www.regentsprep.org/Regents/math/algtrig/ATE6/quadinequalpractice.htm

Graph pic: It was made by me 

The Late BLOG :'(

Aim : How do we use complex conjugates to divide imaginary numbers?

So ... we are on a brand new math journey ....



when dealing with imaginary numbers,things can get a bit complex.

For example we can get a question like =
Express your answer in a, a+ bi form :

3-4i          
3+i

We know,from our previous knowledge that an i in the denominator  wont satisfy the answer because it does not  provide a real answer. To correctly answer this question we need to use the conjugate.

A conjugate, is the opposite of the expression.If the expression has a positive sign we switch it to a negative sign to make it a conjugate.                                 

For Example: 
Initial expression:                                     Conjugated expression:
 24+i                                                                     24-i
 30- i                                                                     30+i

MOVING ON TO THE FIRST EXAMPLE~

Express your answer in a, a+bi form:
3-4i
3+i 

STEP 1: Find the conjugate of the denominator and multiply it to the top and bottom of the expression.
                              
3-4i  =  (3-4i)(3-i)                            
3+i   =  ( 3+i)(3-i)

STEP 2: Distribute the expressions. (try to do one part at a time ) 

(3-4i)(3-i)  =  9-3i-12i+4i2                    (remember the imaginary numbers rule)
                  =  9-15i+4(-1)
                  = 9-15i -4                      (combine like terms)
                  = 5-15i 

(3+i)(3-i)   = 9-3i+3i-i2 
                            = 9-i2                                                           (imaginary number rule)
              = 9-(-1)   (a negative number multiplied by a negative number produces a positive number)
              =9+10
              = 19
Step 3: Combine both answers of the both to have a final answer.
     
 TA-DA YOUR ANSWER:

    5-15i

     19 

your answer could be shown this way as well as :

  5    -  15i

 19       19

Now YOU try it out :)

Simplify:

Choose:  35/37 + (12/37)i  35 + 12i   35/36 + (12/36)i

Simplify:

Choose:  5 - 2i  3 + 2i   15 + 10i

LAST IMAGE:

Works cited:

people walking : http://walkingfit.ucr.edu/images/Men%20and%20Women%20walking_cropped.jpg

Try it problems: http://www.regentsprep.org/Regents/math/algtrig/ATO6/multprac.htm

I love math cup:  http://rlv.zcache.com/math_lover_mug-p168094805368057360bzq92_210.jpg