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\(\text{Rational coefficients means that if $\sqrt{2}+\sqrt{3}$ is a zero of the polynomial then}\\ \text{$-\sqrt{2}+\sqrt{3},~\sqrt{2}-\sqrt{3},$ and $-\sqrt{2}-\sqrt{3}$ must be as well}\\ p(x) = (x-(\sqrt{2}+\sqrt{3})) (x-(-\sqrt{2}+\sqrt{3})) (x-(\sqrt{2}-\sqrt{3})) (x-(-\sqrt{2}-\sqrt{3})) = \\x^4-10 x^2+1\)

\(p = \dfrac{\dbinom{7}{6}\dbinom{5}{4}\dbinom{6}{2}}{\dbinom{18}{12}} = \dfrac{25}{884}\)

We have this equation

.20 x  + .70 ( 30 - x) = .50 (30)    simplify

.20 x  + 21 - .70x   =   15

-.50x  + 21 =  15     subtract  21  from both  sides

-.5x  =  -6    divide  both sides  by - .5

x =  12     [ L  of 20% solution  ]

(30  - x)  =   (30 -12)  =  18  [ L of  70% solution ]

Thanks! : D

Good job, milkcloud.....exactly correct  !!!!!

Angle DXY = 30°

Angle XDY  =   105 °   and   angle DYX  = 45°   and angle  XZY  = 75°

Using the Law of  Sines

XY / sin XDY  =  XD / sin DYX

XY  / sin 105 = 24 / sin 45

XY =  24  sin (105) /sin (45)                   {sin105  = sin  75 }

XY =  24  sin 75   / sin 45

XY   = 24 [ sin (30 + 45) ]/ sin(45)

XY =  24 ( √2)  [ sin 30 cos 45 + sin 45 cos 30 ] 

XY = 24√2 [ (1/2) (1/√2)  +  (1/√2) (√3/2]  

XY = 24√2  [ ( 1 + √3) / [ 2√2]  =   12 (1 + √3)

And  using the Law of Sines again

XY /sin XZY  =  ZY / sin ZXY

12 ( 1 + √3) / sin 75  = ZY / sin 60

12 (1 + √3) /  [  ( 1 + √3) / 2√2) ]  = ZY / [√3/2]

ZY  = (24√2) (√3/2)  =  12√6

So....the area of triangle XYZ  =   (1/2) (XY) (ZY)sin 45  =  (1/2)  ( 12 [ 1 + √3 ]) (12√6)  * 1 / √2  =

72 ( 3 + √3)  units^2  ≈  340.7 units^2

Let's say x is the number of fancy shirts Kim bought, and y is the number of plain shirts she bought. 

27x + 13y = 214

x + y = 10

This means that x = 10 - y. We can plug 10 - y into the equation 27x + 13y = 214. This gives us 27 ( 10 - y ) + 13y = 214.

Expand: 270 - 27y + 13y = 214. Isolate the variable: 27y - 13 y = 270 - 214. Simplify and divide: y = 4. 

To get x, simply minus y (which is 4) from 10. 

Now we know that the number of plain shirts Kim bought is 4, and the number of fancy shirts Kim bought is 6.

Please correct me if I'm wrong! <3

Sorry for my annoyance.

There is no "y" involved...

36x - 8^2.

36 * 3 - 64

108 - 64

44

Let  g =   (x + 6)

Let f =   √x

So

h (x)  = ( f  o g)   = f (g)    =  √ [x + 6 ] 

Quadrilateral PQRS is a square whos side length is 10. Let X and Y be points outside the square so that XQ = YS = 6 and XP = YR = 8. Find XY^2.

SR = 10

YR = 8

YS = 6

Angle SYR = 90°

Angle RSY = ?         tan(RSY) = YR / YS   ∠ RSY = 53.13° 

LY = cos(53.13°) * YS = 3.6

LS = sqrt [(YS)² - (LY)²] = 4.8

YN = SP + 2 * LS = 19.6

NX = SR - 2 * LY = 2.8

XY² = YN² + NX² = 392    

what is the value of 36x-8^2 when x=3 and y=-6   

Is that 36x minus 8 squared? 

When x=3 then           (36 • 3) – (8 • 8) 

                                    108 – 64  =  44 

                                     and it doesn't matter what y equals 

_.

Here :  https://web2.0calc.com/questions/which-pair-of-angle-measurements-are-supplements_1#r1

Since AF  is a perpendicular  bisector.....then by SAS ,   triangle DFA   is congruent to  triangle NFA

So angle  ANF  = angle ADF  =  angle  ADN =  65°

We just need to solve this

-4.9t^2  + 50t  + 8   =   0

Using the  quadratic formula   we have

-50  ±√  [ 50^2 - 4 * (-4.9) (8) ]

____________________________   =  

          2 (-4.9)

-50  + 51.54

___________  =    negative  result

    -9.8

-50 - 51.54

__________  ≈   10.36 sec 

    -9.8  

The  "formula"  for a z score  is

[  given score  - mean score ]  / standard deviation  =  z  score

So....

[ 185 - 193.8 ] / 4.1 ≈  -2.15

Second problem

If  CEF is equilateral, then angle l    = 60 °

And agle C is supplementary  to this angle  = 120°

And angle  (j + k) is vertical to angle  C....so its measure  = 120 °

Supplementary angles always sum to 180 °

A  is  correct

Second one

Angle 1  = Angle  5  [   corresponding angles ] 

Angle 5  + angle 8  =  180     [ supplementary pair ]

120  +  angle 8  = 180      subtract  120 from both sides

angle  8  = 60 

Hi JJ

The area of that whole triangle is    0.5*10*0.2 = 1   

It equals 1!  This is very significant. The area under a probability graph MUST equal 1.

P(3 <  x < 6) 

means what is the probability that the value (x) will lay between 3 and 6.

You can find that by finding the area of the shaded region that is between 3 and 6.

So that would be 

P(3 <  x < 6) =1 - the region in front where x is less than 3 - the region there x is bigger than 6

P(3 <  x < 6) =1 -                     0.5*3*0.1                              -                    0.5*4*0.2

P(3 <  x < 6) =1 -                         0.15                                  -                           0.4

P(3 <  x < 6) =  0.45

OR

 you could have done it as a trapezium, like Geno did.

P(3 <  x < 6) = 0.5(0.1+0.2)*3

P(3 <  x < 6) =0.45

Geno just made a careless error - we all do that sometimes.

I hope that helps. I think these concepts are hard to work out. I remember struggling with them. 

Let the width of the frame  be = W

So....the   area of the picture and frame =  (8 + 2W) (12 + 2W)

And  this  whole area  -  the area of the picture =  the area of the  picture.....so....

(8 + 2W) ( 12 + 2W)  - 96 =  96     

96 + 24W + 16W + 4W^2  - 96 = 96    simplify

4W^2 + 40W  - 96  =  0        divide through by 4

W^2  + 10W  - 24  =  0     factor

(W + 12) ( W - 2)   =0

Setting each factor to  0  and solving for W produces   

W = -12 inches   [ reject]

or

W = 2 inches   [ correct answer ]

If 59%  of the  data lie above, then we must  be at the  (100 -59)  =  41  percentile

The z score associated  with  this will be  negative  .....using a table of z scores.....the z score associated with .41  is  ≈  -0.23

[ 15 - 23.2 ] / 5.6  = z score =  -1.46  

The percentile associated with this z score is  ≈  7.2%   =  7% (rounded)

Let  H  be the height we are  looking for....so we have....

[ H - 60.7 ] / 8.3  ≥  2.75     multiply both sides  by  8.3

H - 60.7 ≥  22.825     add 60.7   to both sides

H ≥  83.525 ≈     84 cm

[ 560 -550] / 45 = .222   

Using a table of z scores....the  % associated with  a z  score  of  .22  ≈ 58.7%

[ 580 - 550 ] / 45 =   30 / 45  =  .66

Again....using a z score table the %  associated with a  z  score of  .66  ≈ 74.5%

So.....the %  scoring  between        560 and 580  ≈ [ 74.5 - 58.7 ] ≈  15.8%  ≈  16%

I meant to put what is p(3 <  x < 6) ?