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Thanks Rom,

I didn't know that, I shall try and remember.   

I mean I did not know this...

Any rational roots are a factor of the constant term divided by a factor of the coefficient of the highest degree term.

how can i use logarithms to base 2?

Click on this arrow.

Let $S$ be the set of numbers of the form \[n(n + 1)(n + 2)(n + 3)(n + 4),\] where $n$ is any positive integer.

The first few terms of $S$ are \begin{align*} 1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 &= 120, \\ 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 &= 720, \\ 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 &= 2520, \end{align*} and so on.

What is the GCD of the elements of $S$?

I assume:

The GCD of the elements of \(S\) is 120.

\(\begin{align*} 1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 &= 120 = 2^3\times3\times5\times7^0\times 11^0\times 13^0\times ... \times p^0 \\ \end{align*}\)

 All other values contain a multiple of 4 and another even number, so at least a \(2^3\)
 All other values contain a multiple of 3, so at least a \(3^1\)
 All other values contain a multiple of 5, so at least a \(5^1\), while there are 5 concecutive numbers.
The gcd is the product of all prime numbers with the smallest exponent. The exponent zero gives the factor 1.

So this is the value of the first therm.

If P and V are the two distinct solutions to the equation $x^2=x+1$, then what is the value of (\P-V)^2

\(x^2=x+1\\ x^2-x-1=0\\ x=-\frac{p}{2}\pm \sqrt{(\frac{p}{2})^2-q}\)

\(x=\frac{1}{2}\pm \sqrt{\frac{1}{4}+1}\\ x=\frac{1}{2}\pm \sqrt{\frac{5}{4}}\\ x=\frac{1}{2}\pm \frac{1}{2}\sqrt{5}\)

\(P=\frac{1}{2}+\frac{1}{2}\sqrt{5}\\ V=\frac{1}{2}-\frac{1}{2}\sqrt{5}\)

\((P-V)^2=((\frac{1}{2}+\frac{1}{2}\sqrt{5})-(\frac{1}{2}-\frac{1}{2}\sqrt{5}))^2\\ {\color{red}(P-V)^2=(2\sqrt{5})^2}\ small\ mistake\ \color{blue}(P-V)^2=(\sqrt{5})^2 \ \)

\((P-V)^2=5\)

Thanks heureka and Allan!

  !

 Continued Fraction Problem

△ABC ~ △DEF   means   "triangle ABC is similar to triangle DEF."

If you take a triangle and multiply each of its sides by some number, you will get a similar triangle.

For example:

If you take  △ABC  and multiply each of its sides by 2, you will get  △DEF.  So  △ABC ~ △DEF

( Also note that the order of the letters in the triangle name matters.

  For example, in this case it is NOT true that  △ABC ~ △FDE )

F = ma

F = 40 kg * .6 m/s^2

  = 24 N

The number x satisfies 5x^2 + 4 = 3x + 9. Find the value of $(10x - 3)^2$.

Guest has given you a very good teaching answer.

And you have responded appropriately.

I am very pleased with both of you.

This is what i suggest.

Find the answer for x - yes I know there will be 2 answers but just leave them with the plus/minus answer. You will not need to seperate them.

You can work out both at onces.

You have x=   [3(+ or -)√109]/10       

now find 10x

then find 10x - 3

Now it is easy to square it.  Both answers will be the same.

how do i know if i plug in  3+√109/10 or  3-√109/10  for 10x

Use the Quadratic Formula to find the value of x in the first part. Then sub that value for x in the second part.

any 1 plz?

4^x + 2^x - 6  = 0   factor  as 

(2^x  + 3) ( 2^x - 2) = 0

Setting the first factor to 0  and solving produces a non-real solution

For the second factor  we have

2^x - 2  = 0

And it's clear that x  = 1

2log ( a - 2b)  = log a + log b

log (a - 2b)^2  =  log (ab)        this implies that

(a - 2b)^2 = ab

a^2 - 4ab + 4b^2  = ab                                        

a^2 - 5ab + 4b^2  =  0       factor                      

(a - 4b) (a - b)  = 0

So  either

a - 4b = 0       or          a - b  =  0

a = 4b                          a  =  b

a/b  = 4                       a/b = 1

But the second solution means that we are taking the log of a negative number at some point

So

a/b = 4

Yes I understand perfectly well.

You have posted no evidence that you have attempted any of these yourself.

The evidence clearly suggests that you are a cheat and a liar.

\(\frac{-7\pm i\sqrt{15}}{8} \)

4x^2 + 7x + k

The product  of the  roots  =   k/4

So  

-7 + i √15               - 7  -  i √15

________    *      ___________    =

      8                           8

49  - i^2 (15)             49  +  15            64            

__________   =       ________  =   _____ =      1

       64                         64                  64            

So

1  = k/4

k =  4       

Can anyone help, still not sure what the answer is

What is it you don't understand?

4. secant  = r/x

r =   √ [ (-9)^2 + (-2)^2 ]  =    √ [ 81 + 4]  =   √85

So

sec  θ =  √85 / -9   =    - √85 / 9

a=-5; b=1;c= (1/a+1/b)*(1/b-1/a);printc, a, b; a++;if(a<-1, goto2, 0);a=-5;b++;if(b<3, goto2, discard=0;
The following are all the possible values:
24 / 25   with a = -5  and b = 1
15 / 16   with a = -4  and b = 1
8 / 9       with a = -3  and b = 1
3 / 4       with a = -2  and b = 1
21 / 100  with a = -5  and b = 2
3 / 16      with a = -4   and b =2
5 / 36      with a = -3   and b =2
0             with a = -2    and b =2

-3 / 4       with a = -1   and b = 2
16 / 225   with a = -5   and b =3
7 / 144     with a = -4   and b =3
0               with a =-3    and b =3
-5 / 36       with a = -2   and b =3
-8 / 9         with a = -1   and b = 3

I'm sorry, but I think you've misunderstood me. These are my friends HW answers and he has asked me for his help. I have already completed all of his HW to the best of my ability. But I've posted these questions to further my knowledge on these topics and also help out a friend. I hope you understand!

4. Two standard six-faced dice are rolled. Cara scores x points if the sum of the numbers rolled is greater than or equal to their product, otherwise Jeremy scores one point. What should be the value of x to make the game fair?

Here is a table   comparing  sums and products

      1     2     3     4    5    6

1    G    G    G    G   G   G

2    G    E    L     L    L    L

3    G    L    L     L     L   L

4    G    L    L     L    L    L

5    G   L     L     L    L    L

6    G   L     L     L    L    L

Cara has a (12/36)  = (1/3) chance of having a greater or equal sum to a respective product

Jeremy has a (2/3) chance  = twice Cara's chance

So

1 (2/3)   =  x (1/3)

(2/3)  = (1/3) x       divide both sides by (1/3)

(2/3) / (1/3)  =  x

(2/1)  =  x  =  2 ⇒   Cara should get 2 points for a winning turn to make the game fair

Can you explain this better? Im not understanding it.

3.)

cos θ  =  x / r    where  x  is the  x-coordinate of  C  and  r  is the radius.

cos θ  =  x / 26

Let  x  and  y  be the coordinates of point  C. We know...

x² + y²  =  26²

x² + 24²  =  26²

x²  =  26² - 24²

x  =  ±√[ 26² - 24² ]

x  =  ±√[ 100 ]

x  =  ± 10

Point C  is in Quadrant II, so the  x-coordinate of  point C  must be negative.

x  =  -10

cos θ  =  x / 26

cos θ  =  -10 / 26

cos θ  =  -5 / 13

Sorry, cannot read your LaTex !.