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Find the number of real roots of 

\(2x^{2001} + 3x^{2000} + 2x^{1999} + 3x^{1998} + \ldots + 2x + 3 = 0\).

I assume:

\(\begin{array}{|rcll|} \hline 2x^{2001} + 3x^{2000} + 2x^{1999} + 3x^{1998} + \ldots + 2x + 3 &=& 0 \\ 3+2x+3x^2+2x^3+\ldots + 3x^{1998}+ 2x^{1999}+ 3x^{2000}+2x^{2001}&=& 0 \\ (3+2x)\left(1+x^2+x^4+x^6+\ldots +x^{1998}+x^{2000} \right) &=& 0 \\\\ \boxed{1+x^2+x^4+x^6+\ldots +x^{1998}+x^{2000} =\dfrac{1-x^{2002}}{^-x^2}} \\\\ (3+2x)\left( \dfrac{1-x^{2002}}{1-x^2} \right) &=& 0 \\\\ (3+2x)\left( \underbrace{\dfrac{1-x^{2002}}{1-x^2}}_{x\neq \pm 1} \right) &=& 0 \\ 3+2x &=& 0 \\ 2x &=& -3\\\\ \mathbf{x} &=& \mathbf{-\dfrac{3}{2}} \\ \hline \end{array}\)

The  real root  is \(\mathbf{x=-\dfrac{3}{2}}\)

The probability is 11/106.

The equation has no real roots, so the answer is 0. 

(a) 101^2 - 97^2 + 93^2 - 89^2 + ... + 5^2 - 1^2 = 101 + 97 + 93 + 89 + ... + 5 + 1 = 1326.

(b) (a + (2n + 1)d)^2 - (a + (2n)d)^2 + ... + (a + d)^2 - a^2 = (a + (2n + 1) d + (a + 2nd + ... + a + d) + a = n(3a + (n + 2)d).

Hi Guest,

here is set   \(\mathbf{\sin(180^\circ-A) =\sin(A)}\)

The area of the octagon is:   A ≈ 32 u²   

Halpp

Okay, I think I see where the  (-2, -2, -1)  came from, but I am still confused about this question and I want to see what others have to say.

t=tapes

c=CD's

t+c=8

12t+18c=126

multiply {c+t=8} by -12 and set up your system of equations

-12t-12c=-96

12t+18c=126

6c=30

c=5

t=3

check:

5+3=8 yes :)

12(3)+18(5)=126 also yes :)

therefore c=5, t=3

Well, my pattern seems to work, but it kind of broke at 5. Will it break at all multiples of 5? Or is there something else?

Based on what I am seeing, it seems as though all times a multiple of 5 appears, then it has a remainder of 1.

Oh! Thanks for catching that error!

How did you get the remainder of 4⁴ when divided by 5 to be 4?

4⁴ = 256

Divide 256 by 5 and you get a quotient of 51 and a remainder of 1.

Please help me

The equation for the sum of the infinite geometric series is S = a1/(1-r).

We know S and the ratio, so we plug in: 25 = a1/(1-(-1/3)), 25 = a1/4/3, 25 = 3a1/4, 100 = 3a1, a1 = 100/3.

The next term of the sequence is 100/3 times -1/3, which is -100/9

Find the surface area of both

I calculate the difference to be 72 in^2

Multiply by 30: 72*30 = 2160

x = 1, 2, -2isqrt(2), 2isqrt(2)

Thanks so much! 

Acellular slime molds do not have cell walls. As a result, the nuclei are combined into one giant clump of nuclei in the cytoplasm. 

13/20 is equal to 78/120, so 78 people have dogs.

Here you have sneakers (x), socks (y) and shirt (z). Each of these can be expressed as a function of the sneakers. For example the shirt (z) is 0.4 times the price (40%) of the sneakers, and so on. Then we have the following:
z=0.4x

y=0.2z

If y is 0.2z then y can also be writen as 0.2*0.4x=0.08x

x+0.4x+0.08x=118.4

1.48x=118.4

x=118.4/1.48

x=80

z=0.4*80

y=0.2(0.4*80) 

Sneakers cost 80, shirt costs 32 and the socks costs 6.4

Cheers.

Since know the number of tickets is 84, the number of children's tickets can expressed as the total number of tickets minus the adult tickets or 84-x where x is the number of adult tickets. Then we have the following equation:
 

105*x+60(84-x)=7155

Factor out the equation to get

105x+60*84-60x=7155

Next up we get x alone on the left side of the equation and get

105x-60x=7155-60*84

45x=7155-5040

45x=2115

x=2115/45

x=47

y=84-47

y=37

You can check this by putting the values into the equation: 47*105+60*37

Cheers

To calculate this you'll just need to multiply each of the exchange rates. So bunnies are worth 0.75 chickens (3/4), chickens are worth 0.4 pigs and pigs are worth 0.67 donkeys. 

Multiply all of these > 0.75*0.4*0.67=0.2

Since we now know a bunniesis worth 0.2 donkeys we then need (1/0.2=5) 5 bunnies to trade for a donkey.

Please type if you need more clarification.

Let x be large and y be medium, we can write the two totals as

6x+2y=86

4x+3y=72.75

Taking the first equation we can rewrite it to find an expression for x.

6x+2y=86  => 6x=86-2y    < Divide this by 6 and we have x=14.333-0,333y.

Now that we have an expression for x in terms of y, we can subsitute in the second equation as follows

4x+3y=72.75

4(14.33-0.33y)+3y=72.75

57.33-1.33y+3y=72.75

3y-1.33y=72.75-57.33

1.67y=15.42 => y=15.42/1.67

y=9.23

x=14.33-0.33*9.23

x=11.26

Note that I've rounded up to two decimals which creates a slight deviation.