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chip       panda

10           0        10 c 0 =      1

9             1        10 c 9 =    10

8             2        10 c 8 =    45

7             3         10 c 7 = 120

6             4         10 c 6 = 210

5             5         10 c 5 = 252

4             6                       210

3             7                       120

2             8                         45

1             9                         10

0            10                          1        

                      total =      1024 ways  

   Ahhh....misread the question.....

Thanx, jugoslav !  

Congratulations ElectricPavlov!!! Your answer is correct!!!

....but how do you know the image is drawn to scale????

5 ft * sin 5^o     * 12 in/ft  = ..........inches

"Factor f(x) into linear factors given that k is a zero of f(x).f(x) = 4x3 + 23x² – 34x +7; k= 1"

What is \(3^{-1}+3^{-2} \pmod {25}\)?
Express your answer as an integer from 0-24 inclusive.

My attempt:

\(\begin{array}{|rcll|} \hline && 3^{-1}+3^{-2} \pmod {25} \\ &\equiv& 3^{-1}+3^{-1}3^{-1} \pmod {25} \\ &\mathbf{\equiv}& \mathbf{3^{-1}\left( 1+3^{-1}\right) \pmod {25}} \\ \hline \end{array} \)

\(3^{-1} \pmod {25} = \ ?\)

\(\begin{array}{|rcll|} \hline 3^{-1} \pmod {25} &\equiv& \dfrac{1}{3} \pmod {25} \\ &\equiv& 3^{\phi(25)-1} \pmod {25} \quad | \quad \phi(25)=20 \\ &\equiv& 3^{20-1} \pmod {25} \\ &\equiv& 3^{19} \pmod {25} \\ &\equiv& 3^{3*6+1} \pmod {25} \\ &\equiv& 3^{3*6}*3 \pmod {25} \\ &\equiv& \left(3^3\right)^6*3 \pmod {25} \quad | \quad 3^3 =27 \equiv 2 \pmod{25} \\ &\equiv& 2^6*3 \pmod {25} \\ &\equiv& 192 \pmod {25} \\ \mathbf{3^{-1} \pmod {25}} &\equiv& \mathbf{17 \pmod {25}} \\ \hline \end{array}\)

\(\begin{array}{|rcll|} \hline 3^{-1}+3^{-2} \pmod {25} &\mathbf{\equiv}& \mathbf{3^{-1}\left( 1+3^{-1}\right) \pmod {25}} \quad | \quad \mathbf{3^{-1} \pmod {25}} \equiv \mathbf{17 \pmod {25}} \\ &\equiv& 17 ( 1+17 ) \pmod {25} \\ &\equiv& 17*18 \pmod {25} \\ &\equiv& 306 \pmod {25} \\ \mathbf{3^{-1}+3^{-2} \pmod {25}} &\equiv& \mathbf{6 \pmod {25}} \\ \hline \end{array}\)

Thanks very much!

\(\displaystyle ab+2a+2b=-16, \\ \text{so}\\ab+2a+2b+4=-12, \\ (a+2)(b+2)=-12 \dots \dots (1)\)

Similarly,

\(\displaystyle (a+2)(c+2)=30 \dots \dots(2) \\ (b+2)(c+2)=-10 \dots \dots(3).\)

Let

 \(\displaystyle a+2=t,\\ \text{then from (1) and (2), }\\b+2=-12/t \\ c+2=30/t.\)

Substitute into (3),

\(\displaystyle (-12/t)(30/t)=-10,\\ t^{2}=36,\\ t = \pm6.\)

 \(\displaystyle \text{If }\quad t=6,\text{ then }\\a=4,\quad b=-4, \quad c=3.\)

\(\displaystyle \text{If }\quad t=-6, \text{then} \\a=-8, \quad b=0,\quad c=-7.\)

y=f(x) has a horizontal asymptote of

Hello freshman!

What is the name of f (x)?

\(y=f(x)=\dfrac{1}{x}\) has a horizontal asymptote of  \(y⇒0\)

| - ∞ < x < 0  and  0 < x < ∞

  !

Like so:

I have drawn that to scale 1:1 and found that angle x is 80 degrees. That proves that the triangle CEO is an isosceles triangle!!! 

Hi Melody...apolgies for confusing you.....thanks for your patience in re-clarifying...such a great help...you are really talented!!

A = 2 + 2B
C = 2B - 7

180 = 2 + 2B + B + 2B - 7

180 = 5B - 5

185 = 5B 

  37 = B 

A = 2 + 2B 

   = 2 + 2(37)

   = 76

C = 2B - 7

    = 2(37) - 7

    = 67

∡A = 76^o , ∡B = 37^o , and ∡C = 67^o

Hope this helps :)

2u + 57 + 2u + 71 = 360

u = 58º

The floor and fractional parts switch, so y + floor(x) = 5.7.

LP=KP or JP

Proof of it being an isosceles triangle besides it just looking like one?

In a certain rectangle, its length is 10 more than its width.  If the area of the rectangle is 264, then what are the dimensions of the rectangle?

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

width =>  x         length =>  x + 10

x (x + 10) = 264

x² + 10x - 264 = 0

x = 12

width = 12         length = 22     

12 * 22 = 264 

The area of the square is NOT 72.

Triangle CEO is an isosceles triangle and angle x = 80 degrees.

The area of the square is 72.

2 / 5 = ? / 25

? = 10

10 = a      ( 10 thousand)

10 = 2x^2 - 31x + 25

   0 = 2x^2 -31x + 15     use Quadratic Formula t find x = 15   or  1/2     months  

Sounds lke you have an equilateral triangle.....all the sides are the same size!      There is no 'smaller' and 'bigger' in this triangle .