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Thanks for the grammar note! I'd definitely rather learn that the easy way than the hard way.

The x value that gives the minimum or maximum value for a quadratic function, 

ax² + bx + c  ,  is         - $\frac{b}{2a}$    .

So for this one...

The x value that gives the minimum or maximum value for

9x² + 18x + 7   is       - $\frac{18}{2(9)}$  =  - 1   .

And in this case,  9  is positive, so the parabola opens upward, and we have a minimum.

Thanks!

I'm glad I was able to help you out a bit!

x^4 - 2x^3 - 103x^2 + 200x + 300

Split    -103x^2   into   -3x^2 - 100x^2    and we have

x^4 - 2x^3 - 3x^2 - 100x^2 + 200x + 300    this factors as

x^2 ( x^2 - 2x - 3) - 100 (x^2 - 2x - 3 )      

(x^2 -100) ( x^2 - 2x - 3)

(x + 10) ( x - 10) (x  - 3) ( x + 1)

Setting each factor to 0 and solving for x, the roots are   x = -10, -1, 3  and 10

We have a binomial probability

The probability is given by :

C (5,4) (1/2)^4 (1/2)  =  5 / 2^5  =   5 / 32

We have one turning point..........the symmetry of the data points indicate that this is a parabola with a vertex at (0,-2)....so.....the degree is 2

I had a history test just about a half hour ago and one of the questions was on the significance of the Louisiana Purchase! I remembered this answer so it helped me out a bit  

 2x^2 + 6x + 11   factor out the 2

2 [ x ^2 + 3x  + 11/2]   

Take 1/2  of 3  =  3/2....square it  = 9/4    ....add it and subtract it inside the parentheses

2 [ x^2 + 3x + 9/4  +  11/2 - 9/4 ]  factor  the first three terms and simplify

2 [ ( x + 3/2)^2  +  22/4 - 9/4]

2 [ (x + 3/2)^2  +  13/4 ]      distribute the 2

2 ( x + 3/2)^2  +  13/2

The guest made a slight error.......we can write this as

2 [ x - (-3/2) ]^2  + 13/2

So....h  = -3/2

1.  We take 

All the possible factors of 18           

_______________________    =   ±9/2 ,  ±1/4 , ±2/3

All the possible factors of 12

2. f(x)=3x^4−x^3−27x^2+9x  = 0

Write as  3x^4 - 27x^2 - x^3 + 9x  = 0

Factors  as

3x^2( x^2 - 9) -  x(x^2 -9)

(x^2 - 9) (3x^2 -x)

( x+ 3) (x - 3) ( x ) (3x - 1)

The zeroes   are    -3, 0 , 1/3 and 3

Hey thanks for the response but that didn't work.  Was there a typo in your answer or is my hw wrong (thats possible, its happened before)

The answer to the  first one sould be  -11

2x^3 +5x^2−x−6

Write 5x^2  as  4x^2 + x^2......and we have.....

2x^3 + 4x^2 + x^2 - x - 6

2x^2 ( x + 2) + ( x + 2)(x - 3)

(x + 2) [ 2x^2 + x - 3]

(x + 2) (2x +3) (x - 1)

x^3−4x^2−3x+18 = 0

Using some synthetic division, we have

3  [   1   - 4      -3      18  ]

               3      -3     -18

       _______________

        1    -1      -6       0

The remaining polynomial  is   x^2  - x  - 6    which factors as  (x - 3) ( x + 2)

So ....   the factored form is  (x - 3) ( x - 3) ( x + 2)  = ( x + 2) ( x - 3)^2 = 0

Complete the square:
2 x^2 + 6 x + 11

Group terms containing x, leaving a placeholder constant:
(2 x^2 + 6 x + __) + 11 - __

(2 x^2 + 6 x + __) = 2 (x^2 + 3 x^2 + __):
2 (x^2 + 3 x + __) + 11 - __

Take one half of the coefficient of x and square it. Then add and subtract, multiplying by the factored constant 2 on the outside.
Insert (3/2)^2 = 9/4 on the inside and subtract (2×9)/4 = 9/2 on the outside:
2 (x^2 + 3 x + 9/4) + 11 - 9/2

11 - 9/2 = 13/2:
2 (x^2 + 3 x + 9/4) + 13/2

x^2 + 3 x + 9/4 = (x + 3/2)^2:
2(x + 3/2)^2 + 13/2         and h=3/2

noun

each of a number of independently variable factors affecting the range of states in which a system may exist, in particular.

PHYSICS

a direction in which independent motion can occur.

CHEMISTRY

each of a number of independent factors required to specify a system at equilibrium.

PATRIOTISM

the measure that you would go to for freedom.

Also remember

$(-2x)^{-2}\,=\,\frac{1}{(2x)^2}\,=\,\frac{1}{(2x)(2x)}\,=\,\frac{1}{2\cdot x\cdot 2\cdot x}\,=\,\frac{1}{2\cdot2\cdot x\cdot x}\,=\,\frac{1}{4x^2}$

Thanks a million!

The Louisiana Purchase added 828,000 square miles to the United States or 530,000,000 acres.

The land was bought for $15 million dollars from France in 1803. $15 million dollars in 1803 is equivalent to $307,391,263.51 dollars today!

It added 13 states to the United States, and if you do the math, each acre was bought for 3 cents each!

12  =  3 * 4

16 =   4 * 4

Since 4 is the only matching factor, 4 is the GCF

Here's another method

Take 16 and subtract 12  = 4

Does 4  divide 16 and 12.....???....yes.....so...it is the GCF

As Melody says.....take my answers at your own peril, Mr. Owl

If we have no restictions,  the number of ways is given by

C ( k + n - 1, n - 1)      here k is the number of balls and n is the number of boxes

C ( 5 + 3 - 1 , 3 - 1)

C ( 7, 2)  = 21 ways

If we have to have at least one ball in each box

C ( k - 1, n - 1)

C ( 5 - 1 , 3 - 1 )

C ( 4, 2)   =  6 ways

BTW ......here's a website that will help you with these type of problems :

http://www.careerbless.com/aptitude/qa/permutations_combinations_imp7.php

Thanks, hectictar.......here's another way, but...it uses some Calculus

xy  = 3/2   →   y = 3 / [ 2x ]    (1)

10x + 3y/5     (2)

Sub (1)  into (2)

10x  +  9 / [10x]     take the derivative of this and set it to 0

10  - (9/10)x^(-2)  = 0   multiply through by x^2

10x^2  - (9/10)  = 0     add 9/10 to both sides

10x^2  =  9/10    multiply through by 10

100x^2  =  9       divide both sides by 100

x^2  =  9 / 100    take the positive root

x = 3/10

And  y  is  found using   xy = 3/2

(3/10) y  =  (3/2)     multiply both sides by 10/3

y  = 5

So....the minimum  is       10(3/10) + 3*5 /5    =   3  +  3  =    6

As requested, we will expand on this math problem.

17 = 10 and 7 put together. We already know what 9 is.

So I have added smiley faces to show the numbers of items  we are counting

= 9 faces.

= 17 smiley faces.

Add that all up and you get 26.

The only way for protons in the nucleus to leave an atom is if the element changes to a new element.

So if you have a chemical reaction and you can change what element an atom is, you are changing the number of protons in the atom.

So I would say, Yes, a proton can react and leave the atom.

6j² - 4j + 12       Factor out a  6 .

=  6(j² - 2/3j + 2)       Add and subtract  (2/6)²

=  6(  j² - 2/3j + (2/6)² + 2 - (2/6)²  )      Factor  j² - 2/3j + (2/6)²  as a perfect square trinomial.

=  6(  (j - 2/6)² + 2 - 4/36  )        Combine  2  and  -4/36 .

=  6(  (j - 1/3)² +  17/9 )             Distribute the  6 .

=  6(j - 1/3)² + 34/3

Now it is in the form  c(j + p)² + q  ,  and  q = 34/3  and  p = -1/3  →  q/p  =  (34/3) / (-1/3)  =  -34

Could you expand more on that answer so that I can understand more? Maybe show the process?

 YAY