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Multiples  of  5   =  [ 5995  - 4000  ]  / 5   + 1  =  400  

Multiples   of 5*3 = 15   =    [  5985 - 4005] /15  + 1   = 132  + 1 =    133

Number (mult of 5  but not a mult of 3)=    400    -  133      =   267

x +  y  = 10    ⇒    y   =  10  -  x

So

x^2   -   (10- x ) ^2  = 56

x^2  - x^2   + 20x  -100   =  56

20x  =  156

x =  156 / 20   =     7.8

y =  10  - 7.8    =  2.2

since in $\triangle ABC$, $\overset{-}{AB}=\overset{-}{AC}$, it means that it is an isosceles triangle, which also means that both angles at the bottom are going to be the same

so we know that $\angle ABC=\angle ACB=50^\circ$

by adding the two you get $50^\circ+50^\circ=100^\circ$

a triangle's interior angle are equal to $180^\circ$

knowing that $ \angle ABC+\angle ACB+ \angle BAC=180^\circ  $, by plugging in what we have we get $ 100^\circ+\angle BAC=180^\circ  $

$  \angle BAC=180^\circ-100^\circ   $

$ \boxed{ \angle BAC=80^\circ } $

7.8 and y = 2.2

the 200 comes from adding the 100 together both there and back

the 4 angles must sum to 360 degrees

q    +    5q+15     +     3q+25      +    3q+70      = 360

12q    + 110 = 360

12 q = 250

q = 20 5/6   degrees

3q+70 = 132.50 degrees         

3x^2  +  72x   +  4

3  ( x^2  + 24x  +  4/3)           complete the square on  x inside the parentheses

3  ( x^2  + 24x  + 144  +  4/3  - 144)

3 [ ( x + 12)^2  +  4/3  - 432/3   ]

3 [  (x + 12)^2  -  428/3 ] 

3(x + 12)^2  -  428

3 ( x -  -12)^2   -  428

(a , h , k)   =  (3 , -12 , -428)

2y  = -x +  8               (1)

-y  = -5x +  1   ⇒    y =  5x  -  1    (2)

Sub (2) into (1)  for  y

2 ( 5x - 1)   =  -x +  8

10x  -  2  = - x +  8

10x + x =    2 +  8

11x   = 10

x  = 10/11

y = 5 ( 10/11)  -  1   =   50/11  -- 11/11  =   39/11

Ordered pair =    ( 10/11 ,  39/11)

Only multiples  of 3.......every  other  "n"  will   ptoduce  either  .3(repeating or .6 ( repeating)

So

( 474/3)    =   158   integers

2x^3  - 7x^2  -  6  =  2

2x^3  - 7x^2  -  8  =   0 

By Vieta.....

The  product of the roots =   -(-8)  / 2   =   8 / 2   =   4

i did not send through :/

x^2 +  5x +  6 <   0

(x + 3) ( x + 2)  <  0

This wil be true  when      -3 < x < -2 

3x^2   +  16x  +  4

3  ( x^2  +  (16/3)x  +  4/3)              complete  the  square inside  the parentheses

3 ( x ^2  +  (16/3)x  + 64/9  + 4/3 - 64/9)

3 [  ( x +  8/3)^2   +  12/9 - 64/9)  ]

3   [  ( x + 8/3)^2  -   52/9  ]  

3 ( x +  8/3)^2  -  52/3

3 ( x -  -8/3)^2  - 52/3

(a , h , k)  =    (3 ,- 8/3 , -52/3 )

we are given $  x^2+5x+6<0 $ right? turn the $<$ into $=$:

$  x^2+5x+6=0 $

again, a binomial in which $  a=1$   ;   $b=5$   ;   $c=6 $

plugging that into $ _1x_2=\frac{-b\pm \sqrt{b^2-4ac}}{2a}  $ we get:

$ _1x_2= \frac{-5\pm \sqrt{5^2-4\cdot \:1\cdot \:6}}{2\cdot \:1}  $

$  _1x_2=  \frac{-5\pm \:\sqrt{25-24} }{2\cdot \:1} $

$   _1x_2= \frac{-5\pm \:1}{2\cdot \:1} $

that gives us:

$ x_1=-2$   and  $x=-3 $

D is on side BC of triangle ABC. We know ABC ~ ACD and angle A = 49 degrees.  If BD = CD, what is angle B in degrees? 

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

After a small correction, this becomes a very challenging question.

The y intercept  is  - 4  and  the  slope  is  1/2

So

(0 , -4)  +  (2  , 1)  =  (2 , - 3)  =  J

(2, -3)  +  ( 2 ,1)   =  ( 4, -2)

(4 , -2)  +  (2, 1)  =  (6 , -1)   =  D

And that's all.....!!!

$  4t^2\le 9t-2 \ \ \ \Rightarrow \ \ \  4t^2+2\le 9t  \ \ \ \Rightarrow \ \ \   4t^2+2-9t\le \:0 $

a trinomial

$a=4$   ;   $b=-9 $   ;    $c=2$

plug that into $ _2t_1=\frac{-b\pm \sqrt{b^2-4ac}}{2a}  $:

$_2t_1=\frac{-\left(-9\right)\pm \sqrt{\left(-9\right)^2-4\cdot \:4\cdot \:2}}{2\cdot \:4}$

$_2t_1=\frac{-\left(-9\right)\pm \sqrt{81-32}}{2\cdot \:4}$

$_2t_1=  =\frac{-\left(-9\right)\pm \:7}{2\cdot \:4}$

$t_1=\frac{9+7}{2\cdot \:4}$   and   $t_2=\frac{9-7}{2\cdot \:4}$

those, respectively give you $   t_1=2  $   and $   t_2=\frac{1}{4} $

thus, we get $  \boxed{\frac{1}{4}\le t\le 2}  $

tricky line

google says there are approx 52 weeks in a year; knowing that we can find how much they are paying in a year:

$1000 \times 52 = 52000$$

we can also approx find how much they are going to pay for 25 years:

$  52000\times 25= 1300000 $$

we know that the compounded interest formula is   $  A=P\left(1+\frac{r}{n}\right)^{n t} $  where $P$ is the ammount that is getting invested, which means we have to solve for it:

$   A=P\left(1+\frac{r}{n}\right)^{n t} \ \ \Rightarrow \ \ \ \ \ P= \frac{A}{\left(1+\frac{r}{n}\right)^{nt}}  $ 

we have $A=1300000$$    ;    $  n=52 $   ;    $  t=25  $   ;   

$r=4\% \ \ \Rightarrow \ \ \ r=0.o4$

just plug and chug at this point:

$P=\frac{1300000}{\left(1+\frac{0.4}{52}\right)^{52\times 25}}$

$  P=\frac{1300000}{(1.0076923)^1300} $

$  P=\frac{1300000}{21199}  $

$ P=61\frac{6861}{21199}  $   or   $ P=61.6861  $

the total of the 10 numbers is: 10(85)=850

subtracting 70 and 78 gives 850-70-78=702

dividing by 8 gives: 87.75

x ( x + 6)   >  16            rearrange as

x^2  +  6x   -  16   >   0       factor as

(x  + 8)  ( x - 2)  >   0

This  will    true  when     x <  -8     and   x  > 2

since there are 9 numbers and choose 3 to be in so we can do 9!/6!=504 but some numbers are repeated in the list which are 5 and 6. 5 is repeated three times and 6 is reapeated twice so we divide by 2! and 3!

504/2/6=42

100^2-100+201=10101

101^2-101+201=10301

10301 is prime so the gcd is 1

for \(\LaTeX\)you can't copy and paste you have to type it in your self

The  value of  what   ????

this is how you find the volume of a cylinder:
$V=\pi r^2h $

in our case, the thing is using the diameter instead of r, which is no big deal ; thus you can write it as :  $\pi \left(\frac{1}{2}d\right)^2h=V$

we want to solve for h:

$\frac{\pi }{4}d^2h=V$
get rid of the fraction form:
$\pi hd^2=4V$
take pi d^2 to the other side:
$h=\frac{4V}{\pi d^2}$
and thats it!