Question

For what values of x and y is the equation (x2 + y2)2 = (x2 – y2)2 + (2xy)2 true?

Answer 1

Hint:
Expand each term, using the identities:
\((a+b)^2=a^2+2ab+b^2\)
and
\((a-b)^2=a^2-2ab+b^2\)
where \(a=x^2, b=y^2\)

Answer 2

\(\large\color{black}{ (x^2 + y^2)^2 = (x^2 – y^2)^2 + (2xy)^2 }\)

\(\large\color{black}{ (x^2 + y^2)^2 \color{red}{-(x^2 – y^2)^2} = (x^2 – y^2)^2\color{red}{-(x^2 – y^2)^2} + (2xy)^2 }\)

\(\large\color{black}{ (x^2 + y^2)^2 - (x^2 – y^2)^2 = (2xy)^2 }\)

then I am using a rule, \(\large\color{blue}{ a^2-b^2=(a-b)(a+b) }\)

\(\large\color{black}{ [(x^2 + y^2) - (x^2 – y^2)][(x^2 + y^2) + (x^2 – y^2)] = (2xy)^2 }\)

\(\large\color{black}{ [x^2 + y^2-x^2 + y^2][x^2 + y^2 + x^2 – y^2] = (2xy)^2 }\)

\(\large\color{black}{ [2 y^2][2x^2 ] = (2xy)^2 }\)

\(\large\color{black}{ (4xy)^2 = (2xy)^2 }\)

Answer 3

please, refresh and sorry for the question marks.

Answer 4

tell me now, what do you think?