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CE = (DC * BC) / sqrt(DC² + BC²)

Thank you for responding! Helped a lot. I also have another question:

Suppose the graph of $y=f(x)$ includes the points $(1,5),$ $(2,3),$ and $(3,1)$.  Based only on this information, there are two points that must be on the graph of $y=f(f(x))$. If we call those points $(a,b)$ and $(c,d),$ what is $ab+cd$?

thanks so much!

Okay, so firstly we need to find an input to substitute for n to take advantage of the information we already have. If we let n=3 and plug in, we get  

f(3)=3f(1) - 2f(2)

From that value, we can plug in the information we had into the equation, giving us

f(3)=3(-1) - 2(3) =>    f(3)= -9

Then, we repeat these steps. We now know the values of f(3), f(2), and f(1). Can you take it from there?

All the best.

BasicMaths

Note  that  since  BC  = AC, then   BA =  18sqrt (2)

Note  that triangle  YBX  is similar  to  triangle CBA 

Therefore

BY  / BX  =  CB  / BA

BY  / 12 =  18 / (18 sqrt (2)

BY /12  =  1/sqrt 2

BY      =  12/  sqrt (2)   =   6 sqrt (2)

So   YA =   BA - BY =   18sqrt 2 -  6 sqrt 2  =  12 sqrt 2

So

BY  :  YA  =   6sqrt (2)   :  12 sqrt (2)  =    1 :  2

Just enter it as   4    divided by   3     times    pi   times   27    =

We can use  something  known as Pick's Theorem  here

Lattice point  =  a point  with integer  coordinates

The  area  is  given by

# Lattice points  in interior of  figure +  ( # lattice points  on boundary)  / 2    -1   =

11  +   4/2   -  1    =

11  + 2   - 1   =

12 units^2

This is a parabola  of  the parent graph  of  x^2   shifted  to the  right  by  4 units and  down  10 units

The downward shift  doesn't  matter.....

The axis of symmetry   is  x    =   4

See here :   https://www.desmos.com/calculator/gtvtqdphlj

Two triangles are congruent if they have the same traced outline, possibly up to rotating and flipping. This is equivalent to having the same set of 3 side lengths.

Total 9

Every perfect square which is an odd number can be represented as the difference of two consecutive perfect squares. The odd numbers less than 100 (which is the square root of 10000) should be the answer. So, the answer is 50

Explanation:

Consider (x+1)^2 and x^2 be two consecutive perfect squares.

Difference of those two:

(x+1)^2 - x^2

= x^2 + 2x + 1 - x^2

= 2x + 1 = is an odd number and should also be a perfect square.

For example: consider an odd perfect square : 25

Which can be written as 2(12)+1here, x=12

25 can be written as (x+1)^2 - x^2

= (12+1)^2 - 12^2

= 13^2 - 12^2

= 169 - 144

= 25

I hope you understood.
 

Note that 1400 = 2^3 * 5^2 * 7.

A fraction (reduced to lowest terms) has a terminating expansion if and only if the only prime factors in the denominator are 2 and 5.

In the case of n/1400, we therefore need n to be divisible by 7.

Hence, there are floor(1000/7) = 142 possible values of n.

I hope this helps!

HOPE THIS HELPS!!

15!   =

1 * 2  * 3  *  4  * 5 * (2*3)  * 7 * 8 * (3 *3) * 10 * 11 * (4*3) * 13 * 14 * (5*3)   =

3 * 3 * 3 * 3 * 3 * 3  *  (all the rest)

 n  =  6 positive integer values

We first determine the largest positive integer value of n such that 3^n | 15!. We determine this by counting the number of factors of 3 in the product. There are 5 multiples of 3 in the product, and there is one extra factor of 3 in 9, so there are a total of 5 + 1 = 6 factors of 3 in the product of the first 15 integers. So, for all n between 1 and 6, inclusive, 3^n is a factor of 15!
so it is 6

The value of k is 1/2.

2a   =  2 (-2, 3)   =  < -4, 6 >

Thank you

-5pi/3   =   - 300°  =   360  - 300 =  60°  =  pi/3    (good  job!!!)

8pi/7  = a litle  more difficult....8pi/7 =  pi +  pi/7

Compare  to  22pi/7  =   14pi/7  +  8pi/7  =   2pi  + 8pi/7

So    8pi/7 =   22pi/7

OMG thx so much Cphill ur the best :DDDDD

1/8 ^(1/3)  =  1/2

(1/8)^(2/3)   =1/4

(1/8) ^ ( 5/3)  =  1/32

So we  have

- (1/2)  +  (1/3) / (1/4)  - (2/9)  / (1/32)   =

-1/2  +  4/3  -  64/9    =

-9/18  + 24/18  -  128 / 18   =

-113/18

BE = 3        EF = 2         FC = 3 

Sorry, but the answer is 20.

3x^2    -4x   -x^2  + 7x  +  8  simplifies  to

2x^2  +  3x  +  8  =   f(k -x)

So

2x^2  +  3x  +  8   =   2(k - x)^2  + 3(k -x)  +  8

2x^2  + 3x +  8 =  2 ( k^2  - 2kx + x^2)  + 3k  -3x   +  8

2x^2  +  3x   =  2k^2  - 4kx  + 2x^2  + 3k  - 3x

2k^2  -4kx  + 3k   -6x=  0

2k  ( k - 2x)  +  3 ( k  - 2x)   = 0

(2k + 3)  ( k  - 2x)  =  0     the  first factor  set = to   0   will give us what we need

2k  +  3   = 0

2k   =  -3

k = -3/2

I'm guessing they would both be 0

It will be the same x-distance on the OTHER side of the vertex (hence symmetry)

39,0