algebra problem

Question

Find the expression that replaces ? so that it is true for all x and y that are greater than 0:

1/x+1/y/x+1/x+1/y/y=(?/xy)^2

Answer 1

No parentheses makes this strange.I simplified this way. 1/x + x/y + 1, then I made a common denominator of xy . The result was:

y/xy + x^2/xy + xy/xy

there fore I came up with

( y + xy + x^2 )/ xy

the ? Is x^2 + xy + y

Very weird

Answer 2

Hi Gin,

Without parentheses, I have to assume the 2nd term is the same as 1/yx and the 4th term is 1/(y²). If that is not the case, the answer below will be incorrect.

Simplifying the left side, 1/x+1/y/x+1/x+1/y/y = 1/x+1/yx+1/x+1/(y²)

= 2/x+1/yx+1/(y²)

= 2y²/xy²+y/y²x+x/(xy²)

= (2y²+y+x)/(xy²)

The right side of the equation can be written as ?²/x²y²

Putting these simplifications together, we are trying to solve the following for ?:

(2y²+y+x)/(xy²) = ?²/x²y²

Multiplying both sides by ,x²y² ,we have x(2y²+y+x) = ?²

The expression asked for is ? = SQRT [x(2y²+y+x)]