Algebra Examples

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

Step 1

Factor $2$ out of $2 x - 2 y$ .

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Step 1.1

Factor $2$ out of $2 x$ .

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

Step 1.2

Factor $2$ out of $- 2 y$ .

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

Step 1.3

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

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

Step 2

To write $\frac{x^{2}}{{(x - y)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3

To write $- \frac{x + y}{2 (x - y)}$ as a fraction with a common denominator, multiply by $\frac{x - y}{x - y}$ .

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

Step 4

Write each expression with a common denominator of $2 {(x - y)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{x^{2}}{{(x - y)}^{2}}$ by $\frac{2}{2}$ .

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

Step 4.2

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

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

Step 4.3

Raise $x - y$ to the power of $1$ .

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

Step 4.4

Raise $x - y$ to the power of $1$ .

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

Step 4.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.6

Add $1$ and $1$ .

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

Step 4.7

Reorder the factors of ${(x - y)}^{2} \cdot 2$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Move $2$ to the left of $x^{2}$ .

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

Step 6.2

Apply the distributive property.

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

Step 6.3

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

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Step 6.3.1

Apply the distributive property.

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

Step 6.3.2

Apply the distributive property.

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

Step 6.3.3

Apply the distributive property.

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

Step 6.4

Simplify and combine like terms.

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Step 6.4.1

Simplify each term.

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Step 6.4.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 6.4.1.1.1

Move $x$ .

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

Step 6.4.1.1.2

Multiply $x$ by $x$ .

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

Step 6.4.1.2

Rewrite using the commutative property of multiplication.

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

Step 6.4.1.3

Multiply $- 1$ by $- 1$ .

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

Step 6.4.1.4

Multiply $x$ by $1$ .

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

Step 6.4.1.5

Rewrite using the commutative property of multiplication.

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

Step 6.4.1.6

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

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Step 6.4.1.6.1

Move $y$ .

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

Step 6.4.1.6.2

Multiply $y$ by $y$ .

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

Step 6.4.1.7

Multiply $- 1$ by $- 1$ .

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

Step 6.4.1.8

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

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

Step 6.4.2

Subtract $y x$ from $x y$ .

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Step 6.4.2.1

Move $y$ .

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

Step 6.4.2.2

Subtract $x y$ from $x y$ .

$\frac{2 x^{2} - x^{2} + 0 + y^{2}}{2 {(x - y)}^{2}}$

Step 6.4.3

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

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

Step 6.5

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

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