Question

Show that (x+y)² − (x −y)² = 4xy for all real numbers x and y.

Answer 1

To show: (x+y)² − (x −y)² = 4xy

Using difference of square identity:

$a^2-b^2=(a-b)(a+b)$

Taking left side

$\Rightarrow (x+y)^2-(x-y)^2$

$a\rightarrow x+y$

$b\rightarrow x-y$

$\Rightarrow (x+y+x-y)(x+y-x+y)$

$\Rightarrow (2x)(2y)$

$\Rightarrow 4xy$

hence proved

$(x+y)^2-(x-y)^2=4xy$