ElectricPavlov

Sorry typo     x, y = 16,74     NOT 10,74 as I summarized earlier

oOOPS......here is how

x+y = 4 (sqrt3)

x*y = 3     so   y = 3/x  (sub this into the first equation)

x + 3/x = 4 sqrt3     Multiply through by 'x'

x^2 + 3 = 4 sqrt 3  x     re-arrange

x^2 - 4 sqrt3  x   + 3 = 0          Now use the quadratic formula to find   2 (sqrt 3)-3   and   2 (sqrt 3) + 3

Quadratic Formula:

\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)

2(sqrt 3) - 3     and     2(sqrt 3) +3

Hope you can get through my scrawl...and it is in a manner to which you are familiar !

Scary, huh?     Probably scares Melody more than me!   

Here is the parabola

=8x^3 * 2^(-15) =1

8x^3 = 1/(2^-15)

8x^3 = 2^(15)

2^3 x^3 = 2^15

x^3 = 2^12            Take cube root of both sides

x = 16

Alright, I re-read your question....  for this to 'BE DEFINED'  ,   the value of

x^2 - 4x+1     HAS to be POSITIVE (greater than 0-- because there is no logs of NEGATIVE numbers)

so

x^2 - 4x +1 > 0

Quadratic formula results in  ZEROES at   x = 2 +-sqrt3

If you graph this fxn you will see a parabola which is POSITIVE outside of these values and NEGATIVE between them

so       2+sqrt3 <   x < 2-sqrt3

The 1/3 full one which is twice as large equals  2/3 of a smaller one

so you have    2/3 + 1/2 = 7/6 of  smaller one of 3 yr old wine

you basically are filling THREE casks (the smaller one PLUS the 2x bigger one)

the part that will be 3 yr old will be 7/6  out of 18/6   (three casks)

(7/6) / (18/6)    = 7/18ths of the new mixture will be 3 yr old wine.

This is not an EQUATION...this is a RELATION.

to solve for x, you need to have an = 'something'    to make it ann equation.     Sorry