CPhill

(a - 4) (b - 4)  =  

(ab)^2 -  4(a +b)  +  16       (1)

By Vieta

The  sum of  the roots   a + b =  -1/7     (2)

And the product of the roots,  ab  =  -5/7 =  -5/7      (3)

Subbing   (2)  and (3) into (1)  we get that

(-5/7)^2  - 4(-1/7)  +  16  = 

25/49  + 4/7  + 16  =

[25  + 28 + 784 ] / 49 

837 / 49

Here's away to prove  this

Connect BD

Since  AB  =AD.....then in triangle DAB angles ADB  and ABD  are  equal =  [180 - 110]/2  =

70/2 =  35°

So angle CDB  =180  -65 - 35 =  80°

And  angle BCD  = 20°

So...in triangle CDB  angle CBD  =   180  - 80 - 20  = 180 - 100  = 80°

But....in triangle  CDB,angles  CBD  are equal

So....this means that  CB = CD

But  since  CB = CD   and AB = AD....then ABCD is a kite  because it has two pairs of equal adjacent sides

(b)      Let x  be the mL of A

Then  100 - x  is the  mL of B

And we have that

.80x  + .30 (100-x)  =  .50(100)   simplify

.80x   + 30  - .30x  =  50

.80x - .30x  =  50  - 30

.50x  =  20       divide both sides by .50

x = 40  = mL of A

100 - 40  =  60  mL  of B

OK....here we go  .....

(a)  Let x  be  the   mL of Solution A that we need

So....we  have  this equation 

.80x  +  .30 (500) = .70 ( 500 + x)    simplify

.80x + 150 = 350  + .70x

.80x - 70x =  350 - 150

.10x=  200    divide both  sides  by  ,10

x =  2000  (mL  of  Solution A  )

Good answer, Cal   !!!

No prob !!!

(a) , (b)   can't answer  these  unless we know  the ml  of Solution B

(c) This is impossible.....the strongest possible  solution would  be  a pure solution of  A =  80%

There's no way we  can get  a solution that is 90% acid

Note  that   ±√500 ≈  ±22.36

So....the  integers  between   -22.36  and + 22.36   that  are  multiples of  7  are  :

-21, -14, -7 , 0 , 7 , 14,  21

Note  that  the inclusion of 0  might seem  puzzling....but note  that  0 / 7  =  0  =  an integer result

So....0 is a multiple of 7

Maybe  this will help  :  https://www.desmos.com/calculator/46w8zosudi

The  domain for cos (x)  is  [ 0 ,pi ]

The range   of cos (x)  = [ -1, 1 ]

Then  the domain for  the  inverse cosine  is  [ -1,1]

And  the  functional values  (the range)  for the  inverse cosine  is   [ 0 , pi ]

Parentheses  do not  include the  endpoint(s)  of an interval

Thus   (0, 3)   include all the real numbers  between  0 an 3, but  not including  0  or  3

Brackets include the endpoints....so  [0,3 ]  include  all real numbers between 0 and 3  including  0 and 3

Note  that  these  could  be combined

So   [0, 3)  includes  0  but  not 3  on the interval

And  (0, 3]   includes 3 but not 0  on the interval

One more note :  infinity  is  never bracketed

See the following pic :

Let's  find AC =  √ (BC^2  -  BA^2)  = √ (15^2  -12^2)  = √ [225 - 144] = √ 81   =  9

Because  ABC   is bisected, we  have the  folllwing relationship :

AD /DC  =  AB/ BC

AD/DC  =  12/15  = 4/5

So

AC has 9 parts  and  AD  is 4 of them....so   AD  = 4

And  DC   = 5

Using the Pythagorean Theorem

BD  = √  (BA^2  + AD^2)    = √ ( 12^2  + 4^2) =  √ 160   = 4√ 10

Note  that the  sine  of  ADB   = sine of  BDC

sin ADB   = AD / DB  = 12 / (4√ 10)  =  3/√ 10

And note that BDC is  obtuse, so the  cosine will be negative

 And   cos  BDC    = -√ [ 1  - ( sin ADB)^2]   =  √ [ 1 -  (9/10)]  =  -  √ ( 1/10)  =  -√ 10/10   

CORRECTED  !!!!