CPhill

Thanks, Daisy !!!.....Just wanted to try this one , too....I like these kinds of problems ....

x^2  - y^2  = -12   and    x +  y  = 6   (1)

(x + y) ( x - y)  =  -12

6 ( x - y)  = -12

x - y  =  -2      (2)

Add (1)   and (2)   and we have that   2x  = 4  ⇒   x  = 2

And y =  2 + y  = 6   ⇒  y  = 4

So  (x ,y)  = ( 2, 4)

y = x^2  - 8   ⇒  y^2  = x^4 - 16x^2 + 64   (1)

y^2  = -5x + 44    (2)

Set (1)  = (2)   and solve for x

x^4 - 16x^2 + 64  = -5x + 44

x^4 - 16x^2  + 5x  + 20  = 0

x^2 (x^2 - 16) + 5( x +  4)  = 0

x^2 (x + 4)(x - 4) + 5 (x + 4)  = 0

(x + 4) [ x^2 ( x - 4) + 5 ]   =  0

( x + 4) [ x^3 - 4x^2 + 5 ] = 0

x = -4  is one solution

Also  x = -1  is another solution

To find the remaining polynomial  use synthetic division

-1 [ 1  - 4    0     5]

           -1    5    -5

   ____________

     1   -5    5     0

So...the remaining polynomial is  x^2 - 5x + 5....set to  0

x^2-5x + 5  = 0

x^2 - 5x + 25/4  = -5 + 25/4

(x - 5/2)^2  = 5/4     take both roots

x - 5/2  = ± √5 /2

x = 5/2 + √5/2    or     5/2 - √5/2

When  x  = -4, y= (-4)^2 - 8   = 8

           x = -1 , y = (-1)^2 - 8  = -7

           x = 5/2 + √5/2,   y = (5//2  + √5/2)^2  - 8  =  (1/2)(5√5 - 1)= (1/2)(√125 - 1)

            x = 5/2  - √5/2, y = (5/2 - √5/2)^2  - 8  = (-1/2)(5√5 + 1)  = (-1/2)(√125 + 1)

So....the product of the y coordinates is

( 8 * -7 * (1/2)*(-1/2)(√125 - 1)(√125 + 1 ] =

[ -56 * (-1/4) * (125 - 1) ] =

[ 14 * 124 ]  =

1736

x^2 + y^2  = 14x + 48y     rearrange

x^2 - 14x + y^2 - 48y  = 0    complete the square on x, y

(x^2  - 14x + 49) + (y^2 - 48y + 576)  =  49 + 576

(x - 7)^2  +  ( y - 24)^2  =  625

(x - 7)^2  + (y - 24)^2  = 25^2

This is a circle  with a center at  ( 7, 24)  and a radius of 25

The minimum y value will occur  at  ( 7, 24 - 25)  = ( 7, -1)

So...the  minimum y value  =  -1

The total arrangements possible  = 8!

Now...let's consider the arrangements possible if all three DO sit together [in consecutive seats ]

Considering them as  a single entity....they can be arranged  as 3!  = 6 ways

And they can be seated in any 6 positions  [ positions 1 - 6 ]  = 6 ways

And for each of these arrangements the other people can be seated in 5!  = 120 ways

So...the total arrangements possible if they DON'T sit together   =

Possible arrangements  - Arrangements where they sit together  =

8!  - [ 6 * 6 * 5!]  =

40320 - 36*120  =

40320   - 4320  =

36000  ways

a^2  - b^2   = 8   and  ab  = 2 

So 

(a^2 - b^2) (a^2 - b^2)  =  (8) * (8)

a^4  - 2(ab)^2 + b^4  = 64

a^4 + b^4  = 64 + 2(ab)^2

a^4 + b^4  =  64 + 2(2)^2

a^4 + b^4  = 64 + 8

a^4 + b^4  = 72

5x + 8y  = -9

In the form  ax + by  = c, the slope of this line  is   -a/b = -5/8

So...a perpendicular line will have a negative reciprocal slope  = - (-8/5)  = 8/5

So......the equation of a line with  a slope of  8/5  passing through (10,10) is

y =(8/5)(x - 10) + 10

y  =(8/5)x - 16 + 10

y = (8/5)x  -  6

So.....m =  8/5    and   b   = -6  =  -30/5

So....m + b   =     8/5  - 30/5  =    -22/5

Maybe an easier way.....but.....

A  = (0, 4)

F = (3, 0)

slope of AF = -4/3

equation of line containing AF    ....   y  = (-4/3)x + 4

E  = ( 0, 2)

B = (7,0)

slope of EB  = -2/7

equation of line containing EB  ...  y = (-2/7)x + 2

Find the x coordinate of D by setting these two functions equal

(-4/3)x + 4  = (-2/7)x + 2

4 - 2 = (-2/7  + 4/3)x

2  = [-6 + 28] /21 x

2 = [22/21]x

42/22   =x

21/11   = x

And the associated y coordinate  is  (-2/7)(21/11) + 2  = 16/11

So  D  = (21/11 , 16,11)

And the equation of the line joining AB  is  y  =   (-4/7)x + 4

Write this is standard form  ....   4x + 7y  - 28   =   0

Now...using the  equation to find the distance from  a point to a line we can find the altitude of triangle ABD

l   4(21/11) + 7(16/11)  - 28  l                   l  -112/11 l              112/11

_______________________   =           __________  =        _____

     √[ 4^2 + 7^2 ]                                         √65                       √65

And considering AB to be the base, its length is  √ [4^2  + 7^2]  = √65

So...the area  of triangle ABD  =  (1/2) √65  * (112/11) / √65  =

(1/2) (112 / 11)  =

112 /22  =

56 /21  units^2

As tertre and Melody said, this will be undefined whenever the denominator   = 0

And the denominator factors as ( x + 2) ( x - 2)

So   either

x +  2   =  0  ⇒  x  = -2

or

x  - 2   = 0 ⇒   x  =  2

So   [ -2,  2  ]  make the expression undefined

So...I assume that we want to know how much paint we need to cover the kitchen wall and the bathroom

We can solve this with a proportion

4 / 120    =     x  /  ( 67 + 53)      where x is the amount of paint we need

4 /120  =  x  / 120

So....it appears that  x   =  exactly 4 quarts

10x^2  + 20x  - 80   take out the  greatest common factor, 10

10  (x^2  + 2x  - 8)

To factor the polynomial, we need two numbers that add to 2  and multiply to  -8.....

4  and -2    seem to be  just right....so we have

10 ( x + 4) ( x - 2)