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-1/2  - √ 3/2  = 

[ -1 - √ 3] / 2  =

- [ 1 + √ 3 ] /  2 ≈   1.366

But √ 2 ≈  1.414

So......these are not equal......

You're right melody, the only fair test is the test where everybody gets the same hat color (not that trivial, but true). But i think this is not the way you were supposed to answer the question. Here's how i think we are supposed to answer it:

Suppose you have a black hat. That means each of the other competitors see one black hat and one white hat. And because the other competitors are known to be very intelligent, they can infer from that situation they have a white hat and say that color.

Why? Each of the other competitors thinks for himself "suppose i had a black hat. That means the othee competitor WITH THE WHITE HAT (not you, we assumed you have a black hat) sees 2 black hats, and because there are only 2 black hat IMMEDIATELY realizes he has a white hat. But that didnt happen, so i have a white hat.

But that didnt happen either, so you have a white hat.

Here's another method to solve this......

Connect  MN......Since, this segment divides the sides AB and BC of triangle ABC proportionally, then MN is parallel to AC

And AM  is a transversal  between two parallel lines so that angles MAC and AMN  form equal alternate interior angles.......and angle APC = angle MPN since they are both vertical angles

So......by angle-angle congruency, triangle APC is similar to triangle MPN

Also, since AB and MN are parallel, then angle NMB = angle ACB  and  angle NBM = angle ABC.....so, again, by AA congruency, triangle NBM is similar to triangle ABC

But BC = 2BM....and since similar triangles are similar in all respects, AC = 2NM

But..triangle APC was shown to be similar to triangle MPN..... so.....if AC = 2MN, then PC = 2PN 

And using the distance formula,  CN  = 6

So.....since PC + PN  = CN....then, by substitution,  2PN + PN = 6

...  so 3PN  = 6

 ....so PN = 2

And PC = 2PN = 2(2) =  4

Here's a pic :

The answer is 0. Why?

a2x is the just: a * 2 * x (a multiplied by 2 and x). Thus, you can arrange a2x as '2ax' or 'ax2' and it would still mean the same thing. Looking at the equation "a2x - ax2". You can rearrange the "a2x" to "ax2". What is the "ax2 - ax2"? 

Another thing to keep in mind:

If a = 2, x= 3

Replace those numbers into the equation: a2x - ax2. It check out.

(2*2*3) - (2*3*2)

12 - 12 = 0

If you're still confused, ask.

Let AM lie on a line

The slope is -6/(1.5 sqrt (3))  =  -4/sqrt (3)

And the equation of this line   y =     -4x/sqrt (3)  + 6

Let CN lie on another line

Ths slope is -3/ 3sqrt (3)  =  -1/sqrt(3)

And the equation of this line is y =  -1x/sqrt(3) + 3

Setting these two lines equal and solving for x will give us the x coordinate for P

 -4x/sqrt (3)  + 6 = -1x/sqrt(3) + 3  rearrange

3  =  [ 4 - 1] x / sqrt (3)

3  =  3 x/ sqrt (3)       multiply both sides by  sqrt (3) / 3

sqrt (3)  = x

Subbing this into the second equation to find the y coordinate of P we have that

y =   -1 sqrt (3)  /sqrt(3) + 3 =  2

So......using the distance formula, CP = 

sqrt [ (3sqrt(3) - sqrt (3))^2  + (2 - 0)^2 ]  = sqrt [ (2sqrt(3))^2  + (2)^2 ] =

sqrt [ 12 + 4]  =

sqrt [16]  =   4

Here's a pic

To do the elimination method, line the two equations up and add them together.

-2x - 5y  =  49

4x + 3y  =  35

We could add them together now, but it won't help us find the value of  x  or  y . So multiply the first equation through by  2  .

-4x - 10y  =  98

Now line this up with the second equation and add them together.

 -4x - 10y  =  98

  4x  + 3y  =  35   

   0   - 7y   =  133

-7y  =  133                Divide both sides of this equation by -7 .

y  =  -19

Plug this value for  y  into either equation to find the value of  x  .

-2x - 5(-19)  =  49

-2x + 95  =  49          Subtract  95  from both sides of this equation.

-2x  =  -46                 Divide both sides of this equation by  -2  .

x  =  23

LOL Melody...I also haven't ever heard " iroc " before....but I figured it had to stand for instantaneous rate of change. Maybe it does mean something else!! When I looked it up, I got "International Race of Champions" 

Hi Ginger,

I think your metaphors could be customized but not to worry,  

You cracked me up with:

"I wasn’t referring to his wife as hippopotamus....

Because of my troll-chimp nature, I added that she is rather hippy"

Thanks for the early morning humour, it is a good way to start the day.      

Added:

Also, if you think Gino's avatar looks like a hipppo, even when little, then perhaps you should go get your eyes tested :)

Solve the following system:
{x + y = 97 | (equation 1)
x - y = 39 | (equation 2)
Subtract equation 1 from equation 2:
{x + y = 97 | (equation 1)
0 x - 2 y = -58 | (equation 2)
Divide equation 2 by -2:
{x + y = 97 | (equation 1)
0 x+y = 29 | (equation 2)
Subtract equation 2 from equation 1:
{x+0 y = 68 | (equation 1)
0 x+y = 29 | (equation 2)
Collect results:
Answer: | x = 68       and         y = 29

Please post your question in the "Questions", NOT in the "Answers" section.

Show using Fermat's Little Theorem that 66013 is not prime.

If: 2^(66,013 - 1) mod (66,013) = 1, it is probable prime, otherwise it is composite.

But, 2^(66,012) mod (66,013) =11,626. Therefore, it is not prime!

Whatever base between 2 and 66,013 you may try, you will get a remainder > 1.

This cannot be simplified any further.

Thanks Hectictar,

Guest,

What the heck is iroc?

YOU ARE TOO LAZY TO EVEN ASK YOUR QUESTION PROPERLY!

I looked IROC up online and this is what I got:

" In 1989 Chevrolet introduced a IROC Z28 WS9 collectors edition in relative small numbers as well. The third-generation Camaro continued through the 1992 model year."

SO YOU WANT A CAR !

WHY DIDN'T YOU JUST SAY SO!

x+y=97; x-y=39 / using elimination method/ SHOW WORK

Let's say...

y  =  -2x² + 4x - 7         The instantaneous rate of change is the slope of this graph at any point.

Take the derivative of both sides to get an equation that tells us the slope at any point.

dy/dx  =  d/dx ( -2x²+ 4x - 7 )

dy/dx  =  d/dx (-2x²)  +  d/dx (4x)  -  d/dx (7)           Use the power rule.

dy/dx  =  -4x + 4                          Plug in  -2  for x.

dy/dx  =  -4(-2) + 4

dy/dx  =  12

To solve for x, you must use the square root and remember to split the answers into 2:
 

To solve for x and y, first solve for a variable and plug it into the other equation. Since you specified solving by substitution, I'll use that method. Here are your 2 equations:

1. \(2x+4y=-32\)

2. \(-3x+y=6\)

I'll solve for y in the second equation because it has a coefficient of 1.

Now that I have solved for y in one equation, substitute y in the other.

Plug x into an equation and solve for y. I'll plug it into equation 2

Therefore, the solution set is (-4,-6)

Solve the following system:
{2 x + 4 y = -32
y - 3 x = 6

In the second equation, look to solve for y:
{2 x + 4 y = -32
y - 3 x = 6

Add 3 x to both sides:
{2 x + 4 y = -32
y = 3 x + 6

Substitute y = 3 x + 6 into the first equation:
{2 x + 4 (3 x + 6) = -32
y = 3 x + 6

2 x + 4 (3 x + 6) = (12 x + 24) + 2 x = 14 x + 24:
{14 x + 24 = -32
y = 3 x + 6

In the first equation, look to solve for x:
{14 x + 24 = -32
y = 3 x + 6

Subtract 24 from both sides:
{14 x = -56
y = 3 x + 6

Divide both sides by 14:
{x = -4
y = 3 x + 6

Substitute x = -4 into the second equation:
Answer: | x = -4         and              y = -6

Actually, I didn’t get it wrong.  My comment has nothing to do with Geno’s avatar. It is simply a metaphor I use when something is about half completed.

“I’ve completed the hippo and I’m working on the potamus,” is my typical reply when a patron asks about the progress on a commissioned portrait. One time I responded this way to a patron who asked about my progress on his wife’s life-size portrait. He was rather offended and I had to explain I wasn’t referring to his wife as hippopotamus.

Because of my troll-chimp nature, I added that she is rather hippy and does have the largest potamus I’ve seen in a long time.  He was awkwardly quiet for a long time, but he didn’t cancel the commission until I said that I bought the economical 37-liter vats of oils.  It’s a pity that some people just do not have a sense of humor.

Geno’s avatar does look somewhat like a hippopotamus when displayed in thumbnail size.  I kinda, sorta figured a blarney bag would come along and point out the obvious error of my ways.  He did. And he says, “All rhinoceroses are endangered.” Why, I had no idea!!! That’s a good thing to know. I’ll cut back on my use of rhinoceros horn. I wouldn’t want to be a contributor to the extinction of rhinoceroses – just blarney bags.

Is there any way u can further simplify that answer

If , find the value of 

f(x) = x^2 - 3x + 4

\(f(\sqrt3 - \sqrt2)\\ =(\sqrt3 - \sqrt2)^2-3(\sqrt3 - \sqrt2)+4\\ =3-2\sqrt6+2-3\sqrt3 +3\sqrt2+4\\ =9-2\sqrt6-3\sqrt3 +3\sqrt2\\\)

Yeah, it's really messy, but I think this is what they meant.

The wording of this question is not complete.

The King has devised a FAIR test. Where each Prince has equal chance of knowing or guessing the correct answer.

The only way that the chance of guessing correctly can be the EQUAL is if all three of them have a white hat on.

Well 72 is 12% of 600  That is true.

But the question really does not make sense.

[(2-8)/(2-5)]+[(4-7)x(3-10)]

\(\begin{array}{|rcll|} \hline && \left(\frac{2-8}{2-5} \right) + (4-7)\times (3-10) \\\\ &=& \left(\frac{-6}{-3} \right) + (-3)\times (-7) \\\\ &=& \left(\frac{6}{3} \right) + 3\times 7 \\\\ &=& 2 + 21 \\\\ &=& 23 \\ \hline \end{array}\)

It's 72. 600x0,12=72