Question

If $x-y=2$ and $xy=3$, what is the value of the expression $(x^2+y^2{)}^2+4x^2{y}^2-4xy(x^2+y^2).$

• A. $24$
• B. $16$
• C. $36$
• D. $9$

Answer 1

The correct option is B $16$

Given the expression
$(x^2+y^2{)}^2+4x^2{y}^2-4xy(x^2+y^2)$
On manipulating the terms of this expression we get,
$\Rightarrow (x^2+y^2{)}^2+(2xy{)}^2-2\times \left(2xy\right)(x^2+y^2)$
On comparing with the identity $(a-b{)}^2=a^2+b^2-2ab$ we get,
$\Rightarrow (x^2+y^2{)}^2+(2xy{)}^2-2\times \left(2xy\right)(x^2+y^2)=\left[\right(x^2+y^2)-2xy{]}^2$

On furthur simplifying we get,
$\Rightarrow \left[\right(x^2+y^2)-2xy{]}^2=[x^2+y^2-2xy{]}^2=\left[\right(x-y{)}^2{]}^2=(x-y{)}^4$
Now, putting value of $x-y=2$ in the simplified equation we get,
$(x-y{)}^4=(2{)}^4=16$