CPhill

Simplify as

x^2 + 10x -   4  =  0

Call the roots a,b

Product of roots  = ab  =   -4

Sum  of roots =   a + b =   -10    square both sides

a^2 + 2ab + b^2  = 100

a^2 + 2(-4) + b^2   =100

a^2 + b^2   - 8  = 100

a^2 + b^2  = 108

Simplify  as

x^2 - mx  + 14 =  0

Factors of 14         Sum of factors =      m

1 14                                                      15

-1 -14                                                  -15

2  7                                                        9

-2  -7                                                     -9

4 values

11x^2  + 17x - 21    =  0

a + b =  -17/11

(a - 4) + (b - 4)  =

(a + b)  -  8  =

-17/11  - 8  =

-105/11

\( 9/(1 - \sqrt[3]{3} - \sqrt[3]{3} + \sqrt[3]{9}) \)

9  / ( 1  - 2(3)^(1/3) + 9^(1/3))  

9 / (1  - 2(3)^(1/3) + (3^2)^(1/3) )

9  / ( 1 - 2(3)^(1/3) + 3^(2/3) ]

9 / ( 1 - 3^(1/3) )^2

Let  (1 - 3)  =  [ 1 - 3^(1/3) ] [ 1 + 3^(1/3) + 3^(2/3) ] 

-2  = [ 1 - 3^(1/3) ] [ 1 + 3^(1/3) + 3^(2/3) ] 

1 / [ 1 -3^(1/3) ]   =   [ 1 + 3^(1/3) + 3^(2/3) ] / -2            square both sides

1 / [ 1 - 3^(1/3)]^2  =  [7 + 5*3^(1/3)  + 3*3^(2/3)] / 4      multiply both sides by 9

9 / [ 1 - 3^(1/3) ] ^2   =   (9/4) [ 7 + 5*3^(1/3) + 3*3^(2/3) ]

Simplify

y = 5x^2 -10x + c

y = 6x^2 - x 

Set the equations equal

6x^2  - x =  5x^2  -10x + c

x^2 + 9x  - c  =  0

One solution

9^2  + 4c   = 0

81  =  -4c

c = -81/4

Two solutions

9^2 + 4c >  0

81 > -4c

c < -81/4

No solutions

9^2  + 4c  < 0

81  < -4c

c > -81/4

a^3/b^3  + b^3/a^3  =  ( a^6 + b^6) / (ab)^3

x^2 + 5x +  46   =   0 

ab = 46

a + b = -5    square both sides

a^2 + 2ab + b^2 =  25

a^2 + 2(46) + b^2  =  25

a^2 + b^2  = -67

a^3 + b^3  =  (a + b) (a^2 + b^2  - ab)  =  (-5)(-67 - 46)  =  565

(a^3 + b^3)^2  = a^6 + 2a^3b^3 + b^3  =

a^6 + 2 (ab)^3 + b^6  =  565^2

a^6 + 2(46)^3 + b^6  = 565^2

a^6  + b^6  =  565^2  - 2(46)^3

a^3/b^3  + b^3/a^3  =  ( a^6 + b^6) / (ab)^3 =  [ 565^2 -2(46)^3 ] / [ 46^3 ] = 

124553 / 97336

a^2/b + b^2/a =  (a^3 + b^3)  (ab)

x^2  - 5x + 3 = 0

ab = 3

a + b = 5    square both sides

a^2 + 2ab + b^2 = 25

a^2 + 2(3) + b^2  = 25

a^2 + b^2  = 19

a^3 + b^3  =  (a + b)(a^2  - ab + b^2)  = (5) ( 19 - 3) =  80

So

a^2/b  + b^2/a =  (a^3 + b^3)  / ( ab)  =  80  / 3

First one

3x^2 - 18x + 13 =  0

ab  = 13/3

a + b =  18/3 =  6

1/a  + 1/b =   (a + b)  / (ab)  =  6 / (13/3)  =  18 / 13

Second one

4x^2 + x + 11  = 0

uv = 11/4

u + v = - 1/4     square both sides

u^2 + 2(uv) + v^2  = 1/16

u^2 + 2(11/4) + v^2  = 1/16

u^2 + v^2   = 1/16 - 22/4

u^2 + v^2  = -87 / 16

u / v  +  v /u   = 

[ u^2 + v^2  ]  / [ uv ]  =    (-87/16) /( 11/4) =   (-87/16)(4/11)  = -87/44

Amt you have at first  =  A

Amt  your brother has at first  = B

After you give him $5

A - 5 =  B + 5

A  = B + 10      (1)

If he gives you $10

A + 10 =  2(B - 10)

(B + 10) + 10  = 2B - 20

B + 20  = 2B - 20

B = 40

He has  $40

You have $50

                            A

                                       8

        B         2       K        3           C

AK =  sqrt (AC^2 - CK^2 )  =  sqrt (8^2 - 3^2 )  = sqrt (55)

AB =  sqrt (BK^2 + AK^2 )

AB = sqrt ( 2^2  + 55)

AB = sqrt [ 59 ] ≈ 7.68