Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Factor $2$ out of $2 x^{2} - 2 y^{2}$ .

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Step 1.1.1

Factor $2$ out of $2 x^{2}$ .

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

Step 1.1.2

Factor $2$ out of $- 2 y^{2}$ .

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

Step 1.1.3

Factor $2$ out of $2 (x^{2}) + 2 (- y^{2})$ .

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

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = y$ .

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

Step 2

Factor using the perfect square rule.

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Step 2.1

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x y = 2 \cdot x \cdot y$

Step 2.2

Rewrite the polynomial.

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

Step 2.3

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = y$ .

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

Step 3

Cancel the common factor of $x + y$ and ${(x + y)}^{2}$ .

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Step 3.1

Factor $x + y$ out of $2 (x + y) (x - y)$ .

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

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $x + y$ out of ${(x + y)}^{2}$ .

$\frac{(x + y) (2 (x - y))}{(x + y) (x + y)}$

Step 3.2.2

Cancel the common factor.

$\frac{(x + y) (2 (x - y))}{(x + y) (x + y)}$

Step 3.2.3

Rewrite the expression.

$\frac{2 (x - y)}{x + y}$