Algebra Examples

$\frac{x}{4 - 2 x} = \frac{y}{2 (x^{2} - 4)}$

Step 1

Rewrite the equation as $\frac{y}{2 (x^{2} - 4)} = \frac{x}{4 - 2 x}$ .

$\frac{y}{2 (x^{2} - 4)} = \frac{x}{4 - 2 x}$

Step 2

Multiply both sides by $2 (x^{2} - 4)$ .

$\frac{y}{2 (x^{2} - 4)} (2 (x^{2} - 4)) = \frac{x}{4 - 2 x} (2 (x^{2} - 4))$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $\frac{y}{2 (x^{2} - 4)} (2 (x^{2} - 4))$ .

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Step 3.1.1.1

Rewrite using the commutative property of multiplication.

$2 \frac{y}{2 (x^{2} - 4)} (x^{2} - 4) = \frac{x}{4 - 2 x} (2 (x^{2} - 4))$

Step 3.1.1.2

Cancel the common factor of $2$ .

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Step 3.1.1.2.1

Cancel the common factor.

$2 \frac{y}{2 (x^{2} - 4)} (x^{2} - 4) = \frac{x}{4 - 2 x} (2 (x^{2} - 4))$

Step 3.1.1.2.2

Rewrite the expression.

$\frac{y}{x^{2} - 4} (x^{2} - 4) = \frac{x}{4 - 2 x} (2 (x^{2} - 4))$

Step 3.1.1.3

Cancel the common factor of $x^{2} - 4$ .

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Step 3.1.1.3.1

Cancel the common factor.

$\frac{y}{x^{2} - 4} (x^{2} - 4) = \frac{x}{4 - 2 x} (2 (x^{2} - 4))$

Step 3.1.1.3.2

Rewrite the expression.

$y = \frac{x}{4 - 2 x} (2 (x^{2} - 4))$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $\frac{x}{4 - 2 x} (2 (x^{2} - 4))$ .

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Step 3.2.1.1

Simplify terms.

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Step 3.2.1.1.1

Rewrite using the commutative property of multiplication.

$y = 2 \frac{x}{4 - 2 x} (x^{2} - 4)$

Step 3.2.1.1.2

Factor $2$ out of $4 - 2 x$ .

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Step 3.2.1.1.2.1

Factor $2$ out of $4$ .

$y = 2 \frac{x}{2 (2) - 2 x} (x^{2} - 4)$

Step 3.2.1.1.2.2

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

$y = 2 \frac{x}{2 (2) + 2 (- x)} (x^{2} - 4)$

Step 3.2.1.1.2.3

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

$y = 2 \frac{x}{2 (2 - x)} (x^{2} - 4)$

Step 3.2.1.1.3

Cancel the common factor of $2$ .

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Step 3.2.1.1.3.1

Cancel the common factor.

$y = 2 \frac{x}{2 (2 - x)} (x^{2} - 4)$

Step 3.2.1.1.3.2

Rewrite the expression.

$y = \frac{x}{2 - x} (x^{2} - 4)$

Step 3.2.1.1.4

Multiply $\frac{x}{2 - x}$ by $x^{2} - 4$ .

$y = \frac{x (x^{2} - 4)}{2 - x}$

Step 3.2.1.2

Simplify the numerator.

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Step 3.2.1.2.1

Rewrite $4$ as $2^{2}$ .

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

Step 3.2.1.2.2

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 $b = 2$ .

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

Step 3.2.1.3

Simplify terms.

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Step 3.2.1.3.1

Cancel the common factor of $x - 2$ and $2 - x$ .

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Step 3.2.1.3.1.1

Factor $- 1$ out of $x$ .

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

Step 3.2.1.3.1.2

Rewrite $- 2$ as $- 1 (2)$ .

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

Step 3.2.1.3.1.3

Factor $- 1$ out of $- 1 (- x) - 1 (2)$ .

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

Step 3.2.1.3.1.4

Reorder terms.

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

Step 3.2.1.3.1.5

Cancel the common factor.

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

Step 3.2.1.3.1.6

Divide $x (x + 2) \cdot (- 1)$ by $1$ .

$y = x (x + 2) \cdot (- 1)$

Step 3.2.1.3.2

Simplify by multiplying through.

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Step 3.2.1.3.2.1

Apply the distributive property.

$y = (x \cdot x + x \cdot 2) \cdot - 1$

Step 3.2.1.3.2.2

Simplify the expression.

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Step 3.2.1.3.2.2.1

Multiply $x$ by $x$ .

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

Step 3.2.1.3.2.2.2

Move $2$ to the left of $x$ .

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

Step 3.2.1.3.2.3

Apply the distributive property.

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

Step 3.2.1.3.2.4

Simplify the expression.

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Step 3.2.1.3.2.4.1

Move $- 1$ to the left of $x^{2}$ .

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

Step 3.2.1.3.2.4.2

Multiply $- 1$ by $2$ .

$y = - 1 \cdot x^{2} - 2 x$

Step 3.2.1.3.3

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

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