here are 2 expressions:-
a: x^2 + x - 3
b: y^3 - 2y + 6
substitute in these numbers to evaluate
the expressions
3, -3, 0.5, 0.35

a: 

1.             x² + x - 3

Everywhere you see an x there, replace 
it by (-3) 

         (-3)² + (-3) - 3

Now (-3)² means to multiply (-3) times 
(-3). That gives +9, so replace the 
(-3)² by 9
      
             9 + (-3) - 3

Now 9 + (-3) gives +6 so replace the 
"9 + (-3)" by 6

                    6 - 3

Now replace 6 - 3 by 3, and that's the 
answer,

                        3

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

2.                 x² + x - 3

Everywhere you see an x there, replace 
it by (0.5) 

         (0.5)² + (0.5) - 3

Now (0.5)² means to multiply (0.5) times
(0.5). That gives +0.25, so replace the 
(0.5)² by 0.25
      
             0.25 + (0.5) - 3

Now 0.25 + (0.5) gives +0.75 so replace 
the "0.25 + (0.5)" by 0.75

                    0.75 - 3

Now replace 0.75 - 3 by -2.25, and that's 
the answer,

                        -2.25

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

3.                 x² + x - 3

Everywhere you see an x there, replace it 
by (0.35) 

         (0.35)² + (0.35) - 3

Now (0.35)² means to multiply (0.35) times
(0.35). That gives +0.1225, so replace the 
(0.35)² by 0.1225
      
             0.1225 + (0.35) - 3

Now 0.1225 + (0.35) gives +0.4725 so replace 
the "0.1225 + (0.35)" by 0.4725

                    0.4725 - 3

Now replace 0.4725 - 3 by -2.5275, and that's 
the answer,

                        -2.5275

========================================================

b: 

1.              y³ - 2y + 6

Everywhere you see a y there, replace it by 
(-3) 

         (-3)³ - 2(-3) + 6

Now (-3)³ means to multiply (-3) times (-3) 
times (-3). That gives
-27, so replace the (-3)³ by -27
      
           -27 - 2(-3) + 6

Now - 2(-3) gives + 6 so replace " - 2(-3) " 
by "+ 6"

               -27 + 6 + 6

Now -27 + 6 gives -21 so replace the 
"-27 + 6" by -21

                   -21 + 6

Now replace -21 + 6 by -15, and that's 
the answer,

                       -15

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

2.                
                  y³ - 2y + 6

Everywhere you see a y there, replace it 
by (0.5) 

           (0.5)³ - 2(0.5) + 6

Now (0.5)³ means to multiply 
(0.5) times (0.5) times (0.5). That gives
+0.125, so replace the (0.5)³ by 0.125
      
             0.125 - 2(0.5) + 6

Now - 2(0.5) equals - 1 so replace the 
" - 2(0.5) " by " - 1 " 

               0.125 - 1 + 6

Now 0.125 - 1 gives -0.875 so replace the 
"0.125 - 1" by -0.875

                    -0.875 + 6

Now replace -0.875 + 6 by 5.125, and that's 
the answer,

                        5.125

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

3.                    y³ - 2y + 6

Everywhere you see an y there, replace it 
by (0.35) 

         (0.35)³ - 2(0.35) + 6

Now (0.35)³ means to multiply (0.35) times 
(0.35) times (0.35). That gives
+0.042875, so replace the 
(0.35)³ by 0.042875
      
             0.042875 - 2(0.35) + 6

Now - 2(0.35) gives - 0.7 so replace the
- 2(0.35) by -0.7

                 0.042875 - 0.7 + 6

Now 0.042875 - 0.7 gives -0.657125 so replace 
the "0.042875 - 0.7" by -0.657125

                  -0.657125 + 6

Now replace -0.657125 + 6 by 5.342875, and 
that's the answer,

                        5.342875

Edwin

2xy + x = 12  Please solve for y. 
Show steps. thank you.

Subtract x from both sides:

2xy + x = 12
    - x      - x
----------------
2xy     = 12 - x

Divide both sides by 2x

   2xy     12 - x
  ----- = --------
    2x       2x

    1
   2xy     12 - x
  ----- = -------
    2x       2x 
     1

           12 - x
      y  = -------
             2x

That's it!  Warning:
Don't do any cancelling
in this stage because the 
numerator contains a 
subtraction!

You can leave the answer
that way. Or else you
can separate it into two
fractions like this
first: 

            12      x
      y  = ---- - ----
            2x     2x
        
And then you can cancel the
2 into the 12 in the first
fraction and cancel the x's
in the second fraction:

             6      1
            12      x
      y  = ---- - ----
            2x     2x 
            1       1

            6     1
      y  = --- - ---
            x     2
 
Either answer is perfectly correct.
They are equivalent even though they
don't look the same.  Use this one

           12 - x
      y  = -------
             2x

if you want y to equal to just one 
fraction. Or use this one

            6     1
      y  = --- - ---
            x     2

if you don't mind having two
separate fractions.  Either way
is correct, just don't try to 
cancel without first separating 
into two fractions.

Edwin

                 x + 3
               -------- = 3
                   2

Put the 3 over 1 so everything will be fractions:

                 x + 3     3
               -------- = ---
                   2       1

The LCD of 2 and 1 is 2.  So multiply both sides by
2 over 1:

          2     x + 3      2     3
         --- · -------- = --- · ---
          1       2        1     1


We cancel the 2's on the left and multiply the 
numerators 2·3, getting 6, and denominators 1·1, 
getting 1 on the right:

          1
          2     x + 3      6     
         --- · -------- = --- 
          1       2        1     
                  1

So we can ignore the 1's on the bottome and just have

                  x + 3 = 6      

which you should have no trouble solving.

Edwin

what set of numbers do: 
pi
0
-35
-31.8 
belong to a piece?

        p -- real, irrational
    0 -- real, rational, integer, whole number
  -35 -- real, rational, integer
-31.8 -- real, rational

Edwin

b to the eighth power

That means 

b times b times b times b times b times b times b times b. 
For example, when b represents 2, then "b to the eighth power",
written more concisely as

b8

means this:

2 times 2 times 2 times 2 times 2 times 2 times 2 times 2,
which comes out to be 256.

And if b represents 3, then "b to the eights power",
or b8 means this:

3 times 3 times 3 times 3 times 3 times 3 times 3 times 3,
which comes out to be 6,561.

| If b equals this | then b8 means this | which comes out to be this |
|                  |                    |                            |         
|         0        |   0×0×0×0×0×0×0×0  |                0           |
|         1        |   1×1×1×1×1×1×1×1  |                1           |
|         2        |   2×2×2×2×2×2×2×2  |              256           |
|         3        |   3×3×3×3×3×3×3×3  |             6561           |
|         4        |   4×4×4×4×4×4×4×4  |            65536           |
|         5        |   5×5×5×5×5×5×5×5  |           390625           |
|         6        |   6×6×6×6×6×6×6×6  |          1679616           |
   etc. etc. etc.

Edwin

Kandi has $2.65 in nickels
and dimes. If she has 40 coins in all,
how many nickels does she have?


COIN equation:         N + D = 40

MONEY equation   .05N + .10D = 2.65

Can you solve that system of equations? 
If not post again.

Answer: N = 23, D = 13

Edwin

problem solving using two variables.

The sum of two integers is 51.
The larger integer is 3 more than twice the smaller 
integer.


Let the larger integer be x
Let the smaller integer bw y

Replace the words "The sum of two integers" by x + y

So now we have 

x + y is 51.
The larger integer is 3 more than twice the smaller 
integer.

Replace "is by an equal sign " = ".

Now we have

x + y = 51.
The larger integer is 3 more than twice the smaller 
integer.

Replace the words "The larger integer" by " x "

Now we have

x + y = 51.
x is 3 more than twice the smaller integer.

Replace the word "is by " = +

x + y = 51.
x = 3 more than twice the smaller integer.

Replace the words "twice the smaller integer" by " 2y ".

Now we have

x + y = 51.
x = 3 more than 2y

To get "3 more than something" write down the "something"
and then add 3 to it.  So we replace "3 more than 2y" by
"2y with 3 added to it", or rather "2y + 3".

x + y = 51
x = 2y + 3

Can you solve that system of equations?  If not post
again.  Answer x = 35, y = 16

That checks because

The sum of 35 and 16 is 51, and
35 is 3 more than 32, which is twice 16.

Edwin

x² - 15 = 2x

Get 0 on the right by adding -2x to both sides.

x² - 15 - 2x = 0

Arrange the left side in descending order of
eponents of x:

We need to factor the left side:

Give the x² the coefficient of 1

1x² - 2x - 15 = 0

Multiply the red 1 by the blue 15, getting 15.

Now think of a pair of positive integers whose product
is the blue 15 and whose difference is the green 2.
I said difference because the last sign (before the
15) is minus. Had it been plus, I would have said "sum".

We think of the integers 3 and 5 because their product is 
the blue 15 and their difference is the green 2.

Now we use the 3 and 5 to rewrite the green -2 as 3 - 5.
So we rewrite 2x as -3x + 5x. That is, 

     x² - 2x - 15 = 0

becomes

x² + 3x - 5x - 15 = 0

Now we factor x out of the first two terms

x(x + 3) - 5x - 15 = 0

and factor -5 out of the last two terms on the
left:

x(x + 3) - 5(x + 3) = 0

Be careful to notice that when we factor a
NEGATIVE number, -5, out of another NEGATIVE
number -15, we get a POSITIVE 3.

Now we have a common factor, which we can
factor out, namely the (x + 3)'s which I 
color red:

x(x + 3) - 5(x + 3) = 0 

Factor out the red parentheses:

(x + 3)(x - 5) = 0

Setting the first factor (x + 3) = 0 gives x = -3

Setting the second factor (x - 5) = 0 gives x = 5

So the solutions are -3 and 5.

Edwin

hi i really need your help...im like stessed out right now!!!!
i need to know the formulas for perimemeter, are and volume of a 
geometric solid. please!!!!!

Sorry, but different geometric figures and solids have different 
formulas for perimeters and volumes respectively, so you'll have 
to tell us which geometric figures and solids you are talking
about.

As far as perimeter is concerned, it may help you to notice that
the word "perimeter" can be written as "peRIMeter" so the word
"RIM" is contained withing the word "perimeter", and the
peRIMeter of something is the distance all the way around its
RIM.

Edwin

I'm having trouble with a word problem. It is a Number Problem, You have $30 in
change in your drawer, consisting of dimes and quarters. write an inequity that
shows the different number of coins in your drawer. I honestly do do not know
even where to begin. Please Help.

There must be some quarters among your $30 in change, for if there were no
quarters, and all were dimes (300 of them), your $30 wouldn't consist of dimes
AND QUARTERS, but only of dimes.

The most dimes you could have would be when you have the least possible number
of quarters.

You can't have just one quarter because that would leave $29.75 to have all in
dimes, and you can't have that because you would need a nickel, but you don't
have any nickels!

But you can have as few as two quarters, because that would leave $29.50 which
you could have with 295 dimes.

So the least number of quarters you could have would be 2 and the most number
of dimes you could have is 295

Therefore if D represents the number of dimes, then D is less than or equal to
295, so the way to write that as an inequality is

               D < 295

and if Q represents the number of quarters, then since there must be at least
two quarters, the way to write that as an inequality is

               Q > 2


Now we'll go the other way:

There must be some dimes among your $30 in change, for if there were no dimes,
and all were quarters (120 of them), your $30 wouldn't consist of dimes and
quarters, but only of quarters.

The most quarters you could have would be when you have the least possible
number of dimes.

You can't have just one dime because that would leave $29.90 and you can't have
that all in quarters. Neither can you have $29.80, nor $29.70, nor $29.60 only
in quarters. However you can have $29.50 with 118 quarters (which means you
must have at least 5 dimes). 

Therefore the least number of dimes you can have is 5 and the most number of
quarters you can have is 118.

Therefore since Q represents the number of quarters, then Q is less than or
equal to 118, so the way to write that as an inequality is

               Q < 118

and since D represents the number of dimes, then since there must be at least 5
dimes, the way to write that as an inequality is

               D > 5

Putting all the inequalities together:

               D < 295
               Q > 2
               Q < 118
               D > 5

Turn the 2nd and 4th inequalities backward, so they'll all be "less than
or equal"'s
               D < 295
               2 < Q
               Q < 118
               5 < D

Now we can combine the 4th and 1st of these with D in 
the middle as

             5 < D < 295
               
We can combine the 2nd and 3rd with Q in the middle as

             2 < Q < 118

Edwin

W(2W - 5) = 180 need to know what W is.

W(2W - 5) = 180 

Distribute to remove the parentheses:

            2W² - 5W = 180

Now get 0 on the right by adding -180 to both sides

      2W² - 5W - 180 = 0

Use the quadratic formula:
                  ______ 
            -b ± Öb²-4ac
        x = —————————————
                2a 

where x = W, a = 2; b = -5; c = -180

                      ________________ 
             -(-5) ± Ö(-5)²-4(2)(-180)
        W = ———————————————————————————
                     2(2) 
                   _______ 
              5 ± Ö25+1440
        W = ————————————————
                  4

                   ____ 
              5 ± Ö1465
        W = ————————————————
                  4

Using a calculator with the +, we get 10.8188296
Using a calculator with the -, we get -8.318829605

Edwin

How can I find the square root of 456 using the 
trial divisor method?

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

Write 456 as 456.000000.  Then group it into pairs
of digits, right and left of the decimal, Consider
the introductory 4 to be the pair of digits 04. Each
SINGLE digit of the answer will be placed over a PAIR
of digits.
           .  
      ---------------
     |04 56. 00 00 00
        
The largest digit that does not exceed the square root
of the digit pair 04 is 2, so write 2 above the pair of 
digits 04. 

       2               
      ---------------
     |04 56. 00 00 00

Square 2, getting 04.  Write it below the pair of digits 04
Draw a line under it

       2   .    
      ---------------
     |04 56. 00 00 00
      04
      ---

Subtract getting 00 then bring down the next pair of digits
56. Draw a vertical line to the left of 00 56

       2   .  
      ---------------
     |04 56. 00 00 00
      04
      ---
     |00 56
            
So far we have 2 in the answer.  Multiply 2 by 20. (Double it
and annex a 0).  This gives 40.  Write the 40 to the left of
the vertical line:

       2   .            
      ---------------
     |04 56. 00 00 00
       4
      ---
   40|00 56

40 is the TRIAL divisor.  Divide 40 into "00 56" (interpreted as
just 56).  It goes 1+ times, so write 1 as the next digit in the 
answer.

       2  1.  
      ---------------
     |04 56. 00 00 00
       4
      ---
   40|00 56
                             
Now to get the ACTUAL divisor, add the 1 to the the trial divisor,
getting 41, and write this actual divisor under the 40 and place
another vertical line to the right of it.


       2  1.  
      ---------------
     | 4 56. 00 00 00
       4
      ---
   40|00 56
   41|   
         
Multiply the 1 times the ACTUAL divisor, getting 41. Write it
as "00 41" and write it beside the vertical line at the bottom
underneath the 00 56. Draw a line underneath and subtract, getting
15.  Then bring down the next pair of digits 00. Then draw a 
vertical line to the left of the 15.     


       2  1.  
      ---------------
     |04 56. 00 00 00
       4
      ---
   40|00 56
   41|00 41
      -----
        |15  00
                   
The digits in the answer so far are 21.  So multiply this by 20
(Double 21, getting 42, then annex a 0 to get 420).  Write 420
left of the vertical line.  420 is the next TRIAL divisor. 

       2  1.    
      ---------------
     |04 56. 00 00 00
       4
      ---
   40|00 56
   41|   41
      -----
    420| 15  00
                 

Divide the TRIAL divisor 420 into "15 00" (interpreted as
1500).  It goes 3+ times, so write 3 as the next digit in the 
answer.

       2  1.  3    
      ---------------
     |04 56. 00 00 00
       4
      ---
   40|00 56
   41|   41
      -----
    420| 15  00


Now to get the next ACTUAL divisor, add the 3 to the the trial divisor,
getting 423, and write this actual divisor under the 420 and place
another vertical line to the right of it.

       2  1.  3  
      ---------------
     |04 56. 00 00 00
       4
      ---
   40|00 56
   41|00 41
      -----
    420| 15  00
    423| 
     
Now multiply the 3 times the ACTUAL divisor, 423, getting 1296, and
write it as "12 96" underneath the "15 00" and subtract, getting 231.
So write this as "02 31", and draw a vertical line left of it 

       2  1.  3  
      ---------------
     |04 56. 00 00 00
       4
      ---
   40|00 56
   41|00 41
      -----
    420| 15  00
    423| 12  69
        -------
        |02  31 

The digits in the answer so far are 213.  So multiply this by 20
(Double 213, getting 426, then annex a 0 to get 4260).  Write 4260
left of the vertical line.  4260 is the next TRIAL divisor. 


       2  1.  3  
      ---------------
     |04 56. 00 00 00
       4
      ---
   40|00 56
   41|00 41
      -----
    420| 15  00
    423| 12  69
        -------
    4260|02  31 00
    4265|

Continue this process as long as you like. Here is the
finished problem taken to the nearest thousandth:

       2  1.  3  5  4
      ---------------
     |04 56. 00 00 00
       4
      ---
   40|00 56
   41|00 41
      -----
    420| 15  00
    423| 12  69
        -------
    4260|02  31 00
    4265|02  13 25
         ---------
       42700|17 75 00
       42704|17 08 16
             --------
                66 84

To get another decimal place annex another "00" pair
to bring down.

Why does it work?  I won't show the whole thing. But
here's an idea to get you started if you are interested.  

(10u + t)² = 100u² + 20tu + t

Notice that 20u is the coefficient of t in the second term.
That is related to why we multiply u by 20 to get the TRIAL 
divisor 20tu.  The t on the end is related to why we must 
add the t to get the ACTUAL divisor.  

Edwin

What is A = a + (n - 1)d and it says 
to solve for n

This is a formula for the nth term, A, 
of an arithmetic sequence with first
term a and common difference d.

To solve it for n:

         A = a + (n - 1)d

         A = a + d(n - 1)

         A = a + dn - d

A - a + d = dn

Divide both sides by d

 A - a + d     dn
----------- = ----
     d         d


               1
 A - a + d     dn
----------- = ----
     d         d
               1


 A - a + d     
----------- = n
     d         

               A - a + d     
          n = ----------- 
                   d  

Edwin

using the quadratic equation x² - 4x - 5 = 0, 
perform the following task: 
solve by factoring,
solve by completing the square,
and solve using the quadratic formula

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

By factoring:

x² - 4x - 5 = 0

(x - 5)(x + 1) = 0

Setting x - 5 = 0 gives x = 5
Setting x + 1 = 0 gives x = -1

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

By completing the square:

Get the constant term, -5, off the left side by
adding +5 to both sides of the equation:

                          x² - 4x = 5

To the side, multiply the coefficient of x, which is
-4, by 1/2, getting -2.  Then square this -2. getting,
(-2)² or 4.  Now add 4 to both sides:

                      x² - 4x + 4 = 5 + 4 

The left side will factor as (x - 2)(x - 2) or (x - 2)².
We combine the numbers on the right as 9

                         (x - 2)² = 9

Now we take square roots of both sides.

                            x - 2 = ±3

                                x = 2 ± 3

Using the +,  x = 2 + 3, or x = 5

Using the -, x = 2 - 3, or x = -1

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

By the quadratic formula:

        x² - 4x - 5 = 0

The quadratic formula is:

       ax² + bx + c = 0 has solutions:
                  ______ 
            -b ± Öb²-4ac
        x = —————————————
                2a 

In this cases a = 1; b = -4; c = -5

                      ______________
             -(-4) ± Ö(-4)²-4(1)(-5)
        x = ——————————————————————————
                     2(1) 
                  _____ 
             4 ± Ö16+20
        x = ————————————————
                  2

                   __ 
              4 ± Ö36
        x = ————————————————
                  2

                    
              4 ± 6
        x = —————————
                2

Using the +,  

              4 + 6
        x = —————————
                2


             10
        x = ————
              2         

        x = 5 
       
Using the -,  

              4 - 6
        x = —————————
                2


             -2
        x = ————
              2         

        x = -1 

Edwin

please help...thankyou 
3)Suppose a baseball is shot up from the ground straight 
up with an initial velocity of 32 feet per second. A 
function can be created by expressing distance above the 
ground, s, as a function of time, t. This function is 
s = -16t² + vOt + sO 
O
[16 represents 1/2g, the gravitational pull due to gravity 
(measured in feet per second²). vO is the initial velocity 
(how hard do you throw the object, measured in feet per second).

a) What is the function that describes this problem?
Answer: 

>>...initial velocity of 32 feet per second...<<

>>...v0 is the initial velocity (how hard do you throw the 
object, measured in feet per second)...<<

Translation:  vO = 32

>>...baseball is shot up from the ground...<<

>>...If you are standing on the ground, then sO = 0...<<

Translation:  sO = 0

>>...This function is s = -16t² + vOt + sO...<<

Translation: s = -16t² + 32t + 0

             s = -16t² + 32t  

Compare that to

          f(x) = ax² + bx

and we see that s = f(x), t = x, a = -16 and b = 32.

The vertex is the point which has coordinates

( -b/(2a), f(-b/(2a)) ) 

( -(32)/(2·-16), f(-b/(2a)) )

( 1, f(1) )

f(1) = -16(1)² + 32(1) = -16 + 32 = 16

So the vertex is the point (1, 16)

This means that the maximum height of 16 feet is obtained when 
the elapsed time = 1 second. 

b)The ball will be how high above the ground after 1 second?

             s = -16t² + 32t  

             s = -16(1)² + 32(1) = -16 + 32 = 16

c)How long will it take to hit the ground?

It is on the ground when s = 0


             s = -16t² + 32t

             0 = -16t² + 32t

    16t² - 32t = 0

Factor out 16t on the left:

    16t(t - 2) = 0
  
setting 16t = 0 gives t = 0

stting (t - 2) = 0 gives t = 2.

So the ball is on the ground at the start, when t = 0, and
again after the ball goes up and falls back down to the ground
2 seconds later.     


d)What is the maximum height of the ball?

             s = -16t² + 32t  

Compare that to

          f(x) = ax² + bx

and we see that s = f(x), t = x, a = -16 and b = 32.

The vertex is the point which has coordinates

( -b/(2a), f(-b/(2a)) ) 

( -(32)/(2·-16), f(-b/(2a)) )

( 1, f(1) )

What time will the maximum height be attained?

f(1) = -16(1)² + 32(1) = -16 + 32 = 16

So the vertex is the point (1, 16)

This means that the maximum height of 16 feet is obtained when 
the elapsed time = 1 second.  Note that this was the same time
as we were asked to use in part (b), and the 16 feet obtained
in part (b) was the actual maximum height.

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

4)John has 300 feet of lumber to frame a rectangular patio
(the perimeter of a rectangle is 2 times length plus 2 
times width). He wants to maximize the area of his patio 
(area of a rectangle is length times width). What should 
the dimensions of the patio be, and show how the maximum 
area of the patio is calculated from the algebraic equation. 
Show clearly the algebraic steps which prove your dimensions 
are the maximum area which can be obtained. Use the vertex 
form to find the maximum area.

Answer: 
Show work in this space.

        _____L______
       |            |
       |            |W
       |____________|

>>...the perimeter of a rectangle is 2 times length 
plus 2 times width...<<

Translation:  P = 2L + 2W 


>>...John has 300 feet of lumber...<<

Translation:  P = 300

So 2L + 2W = 300
        2W = 300 - 2L
         W = 150 - L
>>...He wants to maximize the area of his patio (area of a 
rectangle is length times width)...<<

Translation: He wants to maximize A, where

        A = LW
        
Substitute (150 - L) for W:

        A = L(150 - L)

        A = 150L - L²

Arrange right side in descending powers of L.

        A = -L² + 150L

Compare that to

     f(x) = ax² + bx

and we see that y = f(x), L = x, a = -1 and b = 150.

The vertex is the point which has coordinates

( -b/(2a), f(-b/(2a)) ) 

( -(150)/(2·-1), f(-b/(2a)) )

( 75, f(75) )

f(75) = -1(75)² + 150(75) = -1(5625) + 11250 = 5625

So the vertex is the point (75, 5625)

This means that the maximum area pf 5625 square feet is obtained when 
the length L = 75.  Substituting 75 for L in

         W = 150 - L  
         W = 150 - 75
         W = 75

This means that the maximium area of 5625 square feet is obtained when
the Length and the width are both 75 feet, which means it is square.
My, that's a big patio!

Edwin

Can anyone give me any tips on factoring the greatest common monomial factor
from a polynomial. I dont have a problem to submit. I have a lesson sheet for
my class but the lesson isnt in the text book we use. If anyone can help I
appretiate it.

6x³y - 9x³y²z + 12x²y³z²

We first look for a number we can take out. 
6, 9 and 12 can all three be divided evenly 
by 3, so we can take out a 3. So far,

GCF = 3....

Now we look at the letter x

6x³y - 9x³y²z + 12x²y³z²

The first term contains x³. The second term
contains x³ and the third term contains x².
Thus x appears in ALL factors so we can
take out the smallest power of x. The 
smallest power of x is x², so we can take 
out x².  So far

GCF = 3x²....

Now we look at the letter y

6x³y - 9x³y²z + 12x²y³z²

The first term contains y or y¹. The second
term contains y² and the third term contains y³.
Thus y appears in ALL factors so we can
take out the smallest power of y. The 
smallest power of y is y¹, so we can take 
out y¹, or just y.  So far

GCF = 3x²y....

Now we look at the letter z

6x³y - 9x³y²z + 12x²y³z²

The first term contains no z. Therefore we
cannot take out any power of z, because to
take out a letter, that letter must appear
in ALL the terms, so the final
GCF is

GCF = 3x²y

Now under the original expression

6x³y - 9x³y²z + 12x²y³z²

We write this GCF and a set of parentheses,
long enough to hold 3 terms.

3x²y(                )

We have three blanks to fill.

6x³y - 9x³y²z + 12x²y³z²
3x²y(__ - ____ + ____)

To fill the first blank we ask the question:
What must we multiply 3x²y by to get the 
first term 6x²y in the original?  The answer
is 2x, because the 3 needs to be multiplied by
2 and the x² needs to be multiplied by x, so
3x²y must by multiplied by 2x to get 6x²y 

6x³y - 9x³y²z + 12x²y³z²
3x²y(2x - ____ + _____)

To fill the second blank we ask the question:
What must we multiply 3x²y by to get the 
second term 9x³y²z in the original?  The 
answer is 3x¹y¹z or just 3xyz, because 
the 3 needs to be multiplied by 3 and the
x² needs to be multiplied by x¹, or just
x, and we must get the z, so we see that
3x²y must by multiplied by 3xyz to get 
9x³y²z.

6x³y - 9x³y²z + 12x²y³z²
3x²y(2x - 3xyz + ____)

To fill the third blank we ask the question:
What must we multiply 3x²y by to get the 
third term 12x²y³z² in the original?  The 
answer is 4y²z², because 
the 3 needs to be multiplied by 4. We do
not need any x's. The y needs to be 
multiplied by y², and we must get z²,
so we see that 3x²y must by multiplied by
4y²z² to get 12x²y³z²

6x³y - 9x³y²z + 12x²y³z²
3x²y(2x - 3xyz + 4y²z²)

That's the final result.

Edwin

I am having problems solving:
Is the square root of x^2  an identity (true for all 
values of x)?  Thanks for your time.

Note:  An IDENTITY is an EQUATION.  What you have here,
"the square root of x^2" is not an EQUATION but only an 
EXPRESSION.  To have an equation, you must have the word
"EQUALS" or an = sign somewhere in there.  Sorry we are
unable to help, but if you will resubmit the problem
with with an equal sign somewhere in there, then we can.

Edwin

5[2 - (2x - 4)] = 2(5 - 3x)

We want to remove the innermost grouping symbols, the parentheses.

To get remove the parentheses -(2x - 4) first write it as -1(2x - 4)

5[2 - 1(2x - 4)] = 2(5 - 3x)

We remove the parentheses by using the distributive principle:

   5[2 - 2x + 4] = 10 - 6x

Combine the 2 and the 4

      5[6 - 2x] = 10 - 6x

       30 - 10x = 10 - 6x

Subtract 30 for both sides

           -10x = -20 - 6x

Add 6x to both sides

            -4x = -20

Divide both sides by -4

              x = 5
Edwin

A small firm produces both AM and AM/FM car radios. The AM radios take 15 h to
produce, and the AM/FM radios take 20 h. The number of production hours is
limited to 300 h per week. The plant's capacity is limited to a total of 18
radios per week, and exisitng orders require that at least 4 AM radios and at
least 3 AM/FM radios be produced per week. Write a system of inequalities
representing this situation. Then draw a graph of the feasible region given
these conditions, in which x is the number of AM radios and y the number of
AM/FM radios.

>>>...require that at least 4 AM radios...be produced per week...<<<

Translation: x is greater than or equal to 4

            x > 4

>>>...require that...at least 3 AM/FM radios be produced per week...<<<

Translation: y is greater than or equal to 3

            y > 3

>>>...The plant's capacity is limited to a total of 18 radios per week...<<

Translation: x and y added together can't be more than 18

        x + y > 18

>>>...The AM radios take 15 h [each] to produce...<<

Translation: to produce x AM radios takes 15x hours

>>>...the AM/FM radios take 20 h [each]...<<<

Translation: to produce y AM/FM radios takes 20y hours.

>>>...The number of production hours is limited to 300 h per week...<<

Translation: 15x hours and 20y hours together must be less than or
             equal to 300 hours

             15x + 20y < 300

We have this system of inequalities:

                     x > 4
                     y > 3
                 x + y > 18
             15x + 20y < 300

We now graph the four boundary lines whose equations are the
above inequalities with equal signs replacing to inequality
signs.

                     x = 4
                     y = 3
                 x + y = 18
             15x + 20y = 300

x = 4 is a vertical line 4 units right of the y-axis
y = 3 is a horixontal line 3 units above the x-axis
x + y = 18 has intercepts (18, 0) and (0, 18)
15x + 20y = 300 has intercepts (20, 0) and (0, 15)

Draw the 4 lines
          

The feasible region is supposed to be shaded.  I can't do that on here,
but it's the region which is to the right of the vertical line, above
the horizontal line and below both slanted lines.

Edwin

 
am I correct?
f(x) = x² + 6x + 5
What is the y- intercept of this function? 
My ans is (5,0) 
am I correct?

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

No.  You have it backwards.  It's (0,5).  Remember, it's x that 
you replace by 0 to find the y-intercept.  You do know, don't you, 
that f(x) is exactly the same thing as y? 

In other words there is no difference between

f(x) = x² + 6x + 5 

and

   y = x² + 6x + 5

f(x) is only a different way of writing y. So when you
plug in 0 for x, you get
 
   y = 0² + 6(0) + 5

   y = 5

So it's (0,5) [and NOT (5,0)! That's important!] 

Remember, that x comes before y in the alphabet, and you
can use that fact to remember that what you substitute
for x comes first and what you get for y, comes second
in the ordered pair that represents the point.  It is
important not to get these backward.

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

and what is the zero's of this parabola? 
My ans is (0,-1),(0,-5) 

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

That's wrong, You have those backwards too.
Remember that x comes before y in the alphabet.
It's (x,y), and not (y,x)! So the x-intercepts
are the points (-1,0) and (-5,0).  But that's
not all you are getting wrong. You are calling 
points represented by ordered pair "zeros". 
This is wrong, too. 

Now let's go through this and see if I can
straighten you out.  Your difficulty is not
with the algebra manipulation. You do that 
correctly. Your difficulty is with the words
and what they mean, and with the notation.

The x-intercepts are found by substituting 0 
for f(x) or y, and solving.

You apparently did this correctly:

                   y = x² + 6x + 5
                   0 = x² + 6x + 5

It looks better when you put the = 0 on the right
instead of the left:

             x² + 6x + 5 = 0
 
          (x + 1)(x + 5) = 0
  
Setting x + 1 = 0 gives x = -1
Setting x + 5 = 0 gives x = -5

But you got the coordinates backward.  Notice that 
the value you substituted for y, namely 0,
is supposed to be written SECOND and the x-value is 
written FIRST.  So the x-intercepts are (-1,0) and 
(-5,0). Don't get the x-value and the y-value 
backward.  When you substitute 0 for x, you write 0 
first like this (0,___) and you then fill in the blank 
with what you end up getting for y. And when you 
substitute 0 for y, you write 0 second like this 
(___, 0) and you fill in the blank with what you end 
up getting for x. 

Now also I need to tell you the difference between "zeros" 
and "x-intercepts".  They are certainly related! But a zero
of a function is just a SINGLE NUMBER.  It's not a point 
and it's not an ordered pair like (-1,0) and (-5,0). What 
we call the "zeros" are just the x-coordinates of these 
x-intercept points, not the whole points.  The "zeros" are 
just the numbers -1 and -5, not the x-intercept points
(-1,0) and (-5,0).  You see the difference, right?  Yes,
math is picky!  :-)  

I hope this helps you get these things straight.

Edwin

s = 3t + 1 
find the range of the function if 
the domain is {-3,0,3}

Plug in the -3 for t.

s = 3t + 1
s = 3(-3) + 1
s = -9 + 1
s = -8

So -8 is one member of the range. 

Plug in the 0 for t.

s = 3t + 1
s = 3(0) + 1
s = 0 + 1
s = 1

So 1 is another member of the range.

Plug in the 3 for t.

s = 3t + 1
s = 3(3) + 1
s = 9 + 1
s = 10

So 10 is the last member of the range. 

So the range is {-8, 1, 10}

That wasn't very hard, was it?

Edwin

Driving down a mountain, Tom finds that he has descended 1800 ft
in elevation by the time he is 3.25 mi horizontally away from 
the top of the mountain. Find the slope of his descent to the 
nearest hundredth. I thought that the first step would be to 
create an equation

y= 3.25x + 1800

No that's wrong.

But I need to find the slope, which requires 2 points (y2-y1 over x2-x1). 

OK, we can do it using that formula.

Let's put him at the origin (0,0).  Then the top of the mountain
is 3.25 miles or 3.25×5290 ft or 17160 feet to the right of the origin
and 1800 feet up. This means the top of the mountain is at the point
(17160, 1800).
                                      (17160,1800)   
 (0,0)\
       
He is at the origin (0,0) and the top of the mountain (17160, 1800) is
at the upper right corner of the above graph.

So let's use the formula for slope

     y2 - y1     
m = ---------
     x2 - x1

where (x1,y1) = (0,0) and (x2,y2) = (17160. 1800)

      1900 - 0       1800      15
m = ----------- =  ------- = ----- = .1 approximately 
     17160 - 0      17160     143

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

Note: We could have just used the fact that slope means
"rise over run" where the rise is 1800 feet and the
run is 3.25 miles or 17160 feet. Then

slope = rise/run = 1800/17160 = 15/143 = .1 approximately

Edwin

2x-3=-11-2x. This is what I have done so far.

2x=2x-3=-11-2x+2x

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


When solving an equation, it is always a mistake to put
two equal signs on the same line. Only put ONE equal 
sign to a line.

Rules:
1. To get rid of a term on one side of an equation, add 
its opposite to BOTH sides of the equation

2.  Get rid of all terms on the LEFT side which DO NOT
contain x, and leave all terms which DO contain x.

3.  Get rid of all terms on the RIGHT side which DO 
contain x, and leave all terms which DO NOT contain x.

4.  When only the unknown letter and its 
numerical coefficient, such as 5x or -3y, are on the 
left side, and the variable does not appear on the
right side, then get rid of the coefficient of x by
dividing both sides by that numerical coefficient.;


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

We are given

          2x - 3 = -11

The 2x contains x and is on the LEFT, so by rule 2,
we leave it there.

The -3 does not contain x and is on the LEFT, so by
rule 2, we get rid of it.  To do this we use rule 1
and add its opposite, which is +3, to both sides. So
we write on the second line:

      2x - 3 + 3 = -11 + 3

Then we simplify. The -3 and the 3 add to 0 so we just
have 2x on the left. We combine the -11 and the 3 on
the right and get -8.  So we write on the third line:  

              2x = -8

Now by rule 4, we must get rid of the 2 coefficient by
dividing both sides by 2, so we write on the fourth
line:

             2x     -8
            ———— = ————
              2      2

Now we do some canceling. The 2's cancel on the left.
And the -8 divided by the 2 on the right gives -4

             1       4 
             2x     -8
            ———— = ————
              2      2               
              1      1
 
Notice that the negative sign of the -8 did not cancel 
on the right, so it's still there and we get -4 on the 
right, so we write on the last line:

               x = -4

Edwin

A 6-foot tall man is standing near a tree on level ground(the shadow of the
tree is 18 feet long).  If the man's shadow is 4 feet long, how many feet tall
is the tree? 

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

Man's height IS TO Man's shadow AS Tree's height IS TO Tree's shadow

Let the tree's height be x. Then we have

                   6 IS TO 4 AS x IS TO 18

Replace the "IS TO"'s by " / " and the "AS" by " = "

                     6/4 = x/18

Cross multiply:

                    4x = 6(18)
                    4x = 108
                     x = 27 feet

Edwin

Please use the Gauss-Jordan method to solve system of equations
23.   6x-3y = 1
     -12x-6y= -2
I thought this one is no solution but the answer (back of the book)
is ((23y + 1)/6,y).  Why?

The system you typed has solution (1/6, 0).  Either you mistyped the
problem or the book is wrong.

A similar system that would have the solution you gave would be:

      6x-23y = 1
      -12x+46y= -2



29.  4x + 4y - 4z = 24
     2x -  y +  z = -9
      x - 2y + 3z =  1

[4   4  -4 | 24]
[2  -1   1 | -9]
[1  -2   3 |  1]

Divide row 1 thru by 4

[4   4  -4 | 24]÷(4)
[2  -1   1 | -9]
[1  -2   3 |  1]
 
[1   1  -1 |  6]
[2  -1   1 | -9]
[1  -2   3 |  1]

Get a 0 where the 2 is by multiplying row 1 by -2
and adding it to row 2

[2 -1 1|-9]+(-2)·[1 1 -1|6] = [2 -1 1|-9]+[-2 -2 2|-12] = [0 -3 3|-21] 

[1   1  -1 |   6]
[0  -3   3 | -21]
[1  -2   3 |   1]

Divide row 2 thru by -3

[1   1  -1 |   6]
[0   1  -1 |   7]
[1  -2   3 |   1]

Get a 0 where the 1 is in row 3 column 1 by multiplying 
row 1 by -1 and adding to row 3:

[1 -2 3|1]+(-1)[1 1 -1|6] = [1 -2 3|1]+[-1 -1 1|-6] = [0 -3 4|-5]

[1   1  -1 |   6]
[0   1  -1 |   7]
[0  -3   4 |  -5]

Get a 0 where the -3 is by multiplying row 2 by 3 and adding to
row 3

[0 -3 4|-5]+(3)[0 1 -1|7] = [0 -3 4|-5]+[0 3 -3|21] = [0 0 1|16]

[1   1  -1 |   6]
[0   1  -1 |   7]
[0   0   1 |  16]

This matrix represents this system:

1x + 1y - 1z =  6
0x + 1y - 1z =  7
0x + 0y + 1z = 16

or

 x +  y -  z =  6
      y -  z =  7
           z = 16
 
Now back-substitute:

Substitute 16 for z in the 2nd equation:

       y - z = 7
      y - 16 = 7
           y = 23

Substitute 23 for y and 16 for z in the 1st equation:

   x + y - z = 6
 x + 23 - 16 = 6
       x + 7 = 6
           x = -1

So the solution is (x, y, z) = (-1, 23, 16)

Edwin

find the discriminant , and determine the number of real 
solutions. Then solve 3x^2+2x-6=0

a=3, b = 2, c = -6

b²-4ac = 2²-4(3)(-6) = 4+72 = 76

The discriminant is 76, which is a positive number so
there are 2 real solutions.

If it had been 0, there would have been 1 real solution.

It it had been a negative number, there would have been 0 
(no) real solution.

Edwin

I can't figure this problem out.
3x-2y if x=-2 and y=-6

3x - 2y

In place of the x put (-2),
and in place of the y put (-6).

3(-2) - 2(-6)

Multiply the 3 by the (-2) and get -6, so put
(-6) in place of the 3(-2)

(-6) - 2(-6)

Multiply the 2 by the (-6) and get -12, so put
(-12) in place of the 2(-6)

(-6) - (-12)

Now change the subtraction - (-12) to an
addition " + 12 ".

(-6) + 12

Combine those and get 6

That's the answer, 6.

Edwin

The students found that 0.016 of the teachers were either 
completely brave or fearless. If 480 teachers fell into one 
of these categories,how many teachers were there in all?

Let N = the number of teacher there were in all.

480 teachers fell into one of these categories, so

480 teachers were either completely brave or fearless

0.016 of the teachers were either completely brave or fearless.

Therefore,

480 teachers = 0.016 of the teachers

         480 = 0.016N

Divide both sides by 0.016

       30000 = N

There are 30000 teachers

Edwin