CPhill

x^2  + (-4)^2  = 20

x^2 + 16  =  20

x^2  = 4

x  = 2                x =  -2

A = (2, -4)      B =  (-2 , -2)

AB =   2 - (-2)  =   4

x + y  =  10    square both sides

x^2 + 2xy + y^2 =  100

x^2 + y^2  =  100 - 2xy

64 + xy  = 100 - 2xy

3xy  = 36

xy  = 12

y = 12 / x

x + 12 / x  = 10

x^2 + 12  = 10x

x^2  - 10x  =  -12

x^2 -10x + 25  = -12 + 25

(x - 5)^2  =  13     take  both roots

x - 5  = sqrt (13)                            x  - 5   =  -sqrt 13

x = 5 + sqrt (13)                            x  =  5  - sqrt (13)

y  = 12 / ( 5 + sqrt 13)                   y =  12 / (5 -sqrt 13)

y = [12 (5 -sqrt 13)] / 12                y=  12 (5 + sqrt 13) / ( 12)

y = 5 -sqrt (13)                              y = 5 + sqrt 13

(x, y)   =  ( 5 + sqrt 13 , 5 -sqrt 13 )  and  ( 5 -sqrt 13 , 5 + sqrt 13)

Call his salary,  S

S - (1/5)S - (2/3)S =   (2/15)S =   amount he had after food and  transport

(2/15)S  - (1/2)(2/15)S  =  350

(2/15)S - (1/15)S = 350

(1/15)S = 350

S = 5250

Amt spent on food and  transport  = (13/15) (5250)   = $4550

y = x  slope = 1  

y = 2x   slope =  2

m  =   tan  [   [ arctan (1)  + arctan (2) ] /  2  ]  ≈  1.388

Simplify  as

x^2 + sx  +  100 

To be a square of a binomial the discriminant must  =  0

s^2   -  4(1)(100)  = 0

s^2  =   400    take both roots

s  = -20 , 20

x^2 - 4x + y^2 - 2y  = 0          complete the square on x,y

x^2 - 4x + 4 + y^2 - 2y + 1 =  4 +1

(x- 2)^2 + ( y - 1)^2  =  5

This is a  circle centered at ( 2 , 1)  with a radius of sqrt 5

The largest value of x =  2 + sqrt 5

a + b =   4          square both sides

a^2 + 2ab + b^2 =  16

a^2 + b^2   = 16   - 2ab    (1)

  And we are given that

a^2 + b^2  = 6 + 2ab        (2)

Subtract (1) from ( 2)

0 = -10 + 4ab

10 = 4ab

ab = 10 / 4 =  5/2

a^3 + b^3 =  (a + b)  ( a^2 + b^2  - ab)    = 

(4) ( 6 + 2ab - ab)  =

(4) ( 6 - ab)  =

(4) ( 6  - 5/2)  =

(4) ( 7/2)  = 

  14

\(\cfrac{1}{1 + \cfrac{1}{4 + \cfrac{1}{4 + \cfrac{1}{2}}}} \)

Working  from  the bottom up

4 + 1/2 =  9/2

1/ (9/2)  = 2/9

4 + 2/9  =  38/9

1/ (38/9) =  9/ 38

1 + 9/38 = 47/38

1 / (47/38)  =  38 / 47

625^x  =  9   

(5^4)^x  =  9           take each side to the 3/4   power

(5^4)^(3/4 * x)   = 9^(3/4)

(5^3) ^x   =  9^(3/4)

125^x  =   ⁴√(9^3)

125^x  = ⁴√729

A =  2000 ( 1 + .06/12)^(4 * 12)  =  2000 (1 + .06/12)^48     = $2540.98