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what is -2(bx - 5) = 16    

                                                                –2(bx – 5)  =  16    

                                                                –2bx + 10  =  16    

                                                                –2bx  =  6    

(Assuming you want to solve for x)   

                                                                                6          

                                                                      x  =  ––––  

                                                                              –2b    

                                                                                   3         

                                                                      x  =   – –––  

                                                                                    b    

_.

No positive integer n makes this true...

If n = 3

floor [ sqrt 3 ] + floor [ sqrt (4*3) ] =  1 + 3  = 4

If n = 4

floor [ sqrt 4] + floor [sqrt ( 4 *4) ] =  2 + 4  = 6

What does the notation CC mean ?

Similarly AC, BC ?

Note

f(1)  = floor [  -1/10 ]  =   -1   

f(2)  =  floor[  -4/12 ] =   -1

f(3)  =  floor [  -7 / 14 ]   = -1

f(4)  =  floor [  -10 / 16 ] = -1

.

.

f(10) = floor [-28 /28] = -1 

f(11) = floor [  -31 / 30]  = -2

.

.

f(1000)  = floor [ -2998 / 2008 ] = - 2

We have that the first 10 terms evaluate to  -1    and the  last 990 evaluate to -2

Sum  = 10 (-1) + 990 ( -2)  = -1990

\(\lfloor t\rfloor = 3t + 4/5 \)

By graph, I find one value.....t = -0.6

floor (-0.6)   =   3 (-0.6) + 4/5

-1   = -1.8  +  .8

-1  =  -1

Yes, \(x\) is always an element of \(\{x\}\). However it is important to note that \(\{x\}\) is not an element of \(\{x\}\) but rather an (improper) subset. 

tomtom, you bloody idiot! You have asked plenty of questions already, and a great portion of them deal with quadratics and your ineptness at understanding them. One would expect that if you actually read the solutions and understood them that you would be able to deal with trivialities such as this. But no! You simply copy-paste the answer into the answer box and press 'submit' without giving the slightest flying damn about the time and effort you just wasted.

Granted, these homework simps are egotistical morons, but they are also somewhat competent mathematicians who could probably teach Year 3 math if given a degree...

GA

--..-

\(\frac{-21-\sqrt{201}}{10}\)

5x^2 + 21x  + v  =  0 

The discriminant = 201    ....so...

21^2  - 4 (5) (v)   = 201

441  - 20v  =  201

20 v = 441  - 201

20v =  240

v = 240  /  20   =    12

Rearrange as     x^2 + (4m -2) + (m + 6)   = 0

If we have one real root....the discriminant =  0 

So

(4m - 2)^2  - 4 (1) ( m + 6) = 0

16m^2 - 16m + 4 - 4m - 24  = 0

16m^2 - 20m - 20  =   0      divide through by 4

4m^2  - 5m - 5 =  0

m =  [ 5 + sqrt [ 25 - 4 (4)(-5) ] ] / (2 * 4 )  =   [  5 + sqrt ( 105)   ]  /  8

          A

                  E

B        D             C

          12

Since  AB = AC, the triangle is isoceles

AB bisects  BC  so that DC  = 6

And

[ABC ] = (1/2) (BC) (AD)

180  = (1/2) ( BC) (AD)

180 =  (1/2) (12) (AD)

180  = 6 * AD

180 / 6 = AD =  30

And angle ADC  = angle BEC  = 90°

And angle DCA  = angle ECB

So  triangle CDA   is similar to triangle CEB

So   CD  / AD   = CE / BE

So    6/30  =  CE / BE

So     1/5  = CE / BE

5CE  = BE

And

BC^2  =  CE^2 + BE^2

12^2  =   CE^2 + (5CE)^2

144 =      CE^2 + 25CE^2

144  =   26CE^2

144/26 = CE^2

72/13 = CE^2

CE =  sqrt [ 72/13] =  (6 sqrt (2)) / sqrt (13)

tan (angle ACD)  = AD / DC  =  30/6 =  5/1

sin (ACD)  = sin (BCE) = 5  /sqrt (5^2 + 1^2)  =  5/sqrt (26)

Area of triangle  ECD  = (1/2)(DC) (CE)  = (1/2)(6)(6sqrt (2)  /sqrt (13) )  * (5 /sqrt (26))   =

90sqrt (2) / sqrt (13) / [ sqrt (2) * sqrt (13) ]   = 90  / 13

[ ABDE ]  = [ ABC ] - [ ECD ]  =   (180 )  - ( 90/13 )    = 2250 / 13

sorry thats incorrect :((

               A

                     E

B            D             C

If E is  on AC....BE can't be parallel to  AC  !!!

Let XQ =  x     and let PY  =  y

By the interesecting chord theorem we have two equations

PQ * QC =  XQ * QY

and

AP * PB  = PY * XP

So....subbing we have

11*8 = x * (25 + y)

7*4  =  y * (25 + x)             simplify these

88  = 25x + xy        (1)

28  = 25y  + xy        (2)

Subtract  (2)  from (1)

60  = 25 ( x - y)

60 / 25 = x - y

12/5  = x - y

2.4 + y  = x

28 = y * ( 25 + 2.4 + y)

28  = y * ( 27.4 + y)

28  = 27.4y + y^2

y^2 + 27.4y - 28 = 0

Solving this for positive y we have that y ≈ .986

And x = 2.4 + .986 ≈  3.386

XY  ≈ .986 + 25 + 3.386   ≈  29.372

              C

                 5  

                    B                

            4         3

A           D             E

AE  =  sqrt  [ AB^2 + BE^2]  = sqrt [ 4^2 + 3^2 ]  =  sqrt [25]  =  5

AC =  sqrt [ BC^2 + AB^2]  = sqrt [ 5^2 + 4^2]  = sqrt [41]

Let  DE =  x    and  AD = 5 - x

CD  = sqrt [ (sqrt 41)^2  - (5 - x)^2 ]

CD  = sqrt [ 8^2 - x^2 ]

Equate  these

sqrt [  (sqrt 41)^2  - (5 -x)^2]  = sqrt [ 8^2 - x^2 ]         square both sides

(sqrt 41)^2  - (5 -x)^2  = 8^2 - x^2

41 - x^2 + 10x - 25  =  64  - x^2

10x + 16  = 64

10x  =  48

x = 48/10  =   4.8  = DE

I believe that   B - A =  R    and B + A = S

cos A = ( 12/13)       sin A = (5/13)    (A is  a first quad angle)

cos B = (-4/5)          sin B = (3/5)      (B is a second quad angle)

 cos  ( B - A)  =  cos A cosB  + sinAsinB  = ( 12/13)(-4/5) + ( 5/13)(3/5)  =  [-48 + 15 ] / 65  =  -33/65

So  (x, y)  = ( r1, r2)

r1  =   -33       r2  =  sqrt [ 65^2 - 33^2 ]  =   56

So    (r1, r2)  =  ( -33, 56)

Likewise   

cos ( A + B)  =  cosAcosB  - sinAsinB   =    (12/13) (-4/5) - (5/13)(3/5)  =  [-48 - 15] / 65  = -63/65

And again   (x, y) = (s1, s2)

s1 = -63        s2 =  sqrt ( 65^2  - 63^2)  =  16

So

(s1,s2) =  (-63 , 16)

Note that       1/ ( a - 1)  +  1/(b -1)    can  be  simplified to

[ (a + b)  - 2 ]   / ( ab  - (a + b) + 1 )       .......    (1)

Simplify the equation as

3x^2 - 11x  - 7  =  0

By Vieta   the sum of the  roots  (a + b)  =  11/3

And the  product of  the  roots  ab   = -7/3

Subbing into (1) we have

[ 11/3 - 2 ]  /  [ -7/3 - ( 11/3) + 1 ]  =

[  5/3 ] / [ -5 ] =

-5/15  =

-1/3