Simplify (x-y)^2-(x+y)^2

${(x - y)}^{2} - {(x + y)}^{2}$

Step 1

Simplify each term.

Rewrite ${(x - y)}^{2}$ as $(x - y) (x - y)$ .

$(x - y) (x - y) - {(x + y)}^{2}$

Expand $(x - y) (x - y)$ using the FOIL Method.

Apply the distributive property.

$x (x - y) - y (x - y) - {(x + y)}^{2}$

Apply the distributive property.

$x \cdot x + x (- y) - y (x - y) - {(x + y)}^{2}$

Apply the distributive property.

$x \cdot x + x (- y) - y x - y (- y) - {(x + y)}^{2}$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + x (- y) - y x - y (- y) - {(x + y)}^{2}$

Rewrite using the commutative property of multiplication.

$x^{2} - x y - y x - y (- y) - {(x + y)}^{2}$

Rewrite using the commutative property of multiplication.

$x^{2} - x y - y x - 1 \cdot - 1 y \cdot y - {(x + y)}^{2}$

Multiply $y$ by $y$ by adding the exponents.

Move $y$ .

$x^{2} - x y - y x - 1 \cdot - 1 (y \cdot y) - {(x + y)}^{2}$

Multiply $y$ by $y$ .

$x^{2} - x y - y x - 1 \cdot - 1 y^{2} - {(x + y)}^{2}$

Multiply $- 1$ by $- 1$ .

$x^{2} - x y - y x + 1 y^{2} - {(x + y)}^{2}$

Multiply $y^{2}$ by $1$ .

$x^{2} - x y - y x + y^{2} - {(x + y)}^{2}$

Subtract $y x$ from $- x y$ .

Move $y$ .

$x^{2} - x y - 1 x y + y^{2} - {(x + y)}^{2}$

Subtract $x y$ from $- x y$ .

$x^{2} - 2 x y + y^{2} - {(x + y)}^{2}$

Rewrite ${(x + y)}^{2}$ as $(x + y) (x + y)$ .

$x^{2} - 2 x y + y^{2} - ((x + y) (x + y))$

Expand $(x + y) (x + y)$ using the FOIL Method.

Apply the distributive property.

$x^{2} - 2 x y + y^{2} - (x (x + y) + y (x + y))$

Apply the distributive property.

$x^{2} - 2 x y + y^{2} - (x \cdot x + x y + y (x + y))$

Apply the distributive property.

$x^{2} - 2 x y + y^{2} - (x \cdot x + x y + y x + y \cdot y)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} - 2 x y + y^{2} - (x^{2} + x y + y x + y \cdot y)$

Multiply $y$ by $y$ .

$x^{2} - 2 x y + y^{2} - (x^{2} + x y + y x + y^{2})$

Add $x y$ and $y x$ .

Reorder $y$ and $x$ .

$x^{2} - 2 x y + y^{2} - (x^{2} + x y + x y + y^{2})$

Add $x y$ and $x y$ .

$x^{2} - 2 x y + y^{2} - (x^{2} + 2 x y + y^{2})$

Apply the distributive property.

$x^{2} - 2 x y + y^{2} - x^{2} - (2 x y) - y^{2}$

Multiply $2$ by $- 1$ .

$x^{2} - 2 x y + y^{2} - x^{2} - 2 x y - y^{2}$

Step 2

Simplify by adding terms.

Combine the opposite terms in $x^{2} - 2 x y + y^{2} - x^{2} - 2 x y - y^{2}$ .

Subtract $x^{2}$ from $x^{2}$ .

$- 2 x y + y^{2} + 0 - 2 x y - y^{2}$

Add $- 2 x y + y^{2}$ and $0$ .

$- 2 x y + y^{2} - 2 x y - y^{2}$

Subtract $y^{2}$ from $y^{2}$ .

$- 2 x y - 2 x y + 0$

Add $- 2 x y - 2 x y$ and $0$ .

$- 2 x y - 2 x y$

Subtract $2 x y$ from $- 2 x y$ .

$- 4 x y$

Simplify (x-y)^2-(x+y)^2

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