Algebra Examples

$\frac{3}{x + 2} + \frac{2}{x} = \frac{4 x - 4}{x^{2} - 4}$

Step 1

Subtract $\frac{4 x - 4}{x^{2} - 4}$ from both sides of the equation.

$\frac{3}{x + 2} + \frac{2}{x} - \frac{4 x - 4}{x^{2} - 4} = 0$

Step 2

Simplify $\frac{3}{x + 2} + \frac{2}{x} - \frac{4 x - 4}{x^{2} - 4}$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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Step 2.1.1.1

Factor $4$ out of $4 x$ .

$\frac{3}{x + 2} + \frac{2}{x} - \frac{4 (x) - 4}{x^{2} - 4} = 0$

Step 2.1.1.2

Factor $4$ out of $- 4$ .

$\frac{3}{x + 2} + \frac{2}{x} - \frac{4 (x) + 4 (- 1)}{x^{2} - 4} = 0$

Step 2.1.1.3

Factor $4$ out of $4 (x) + 4 (- 1)$ .

$\frac{3}{x + 2} + \frac{2}{x} - \frac{4 (x - 1)}{x^{2} - 4} = 0$

Step 2.1.2

Simplify the denominator.

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Step 2.1.2.1

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

$\frac{3}{x + 2} + \frac{2}{x} - \frac{4 (x - 1)}{x^{2} - 2^{2}} = 0$

Step 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 $b = 2$ .

$\frac{3}{x + 2} + \frac{2}{x} - \frac{4 (x - 1)}{(x + 2) (x - 2)} = 0$

Step 2.2

To write $\frac{3}{x + 2}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{3}{x + 2} \cdot \frac{x}{x} + \frac{2}{x} - \frac{4 (x - 1)}{(x + 2) (x - 2)} = 0$

Step 2.3

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

$\frac{3}{x + 2} \cdot \frac{x}{x} + \frac{2}{x} \cdot \frac{x + 2}{x + 2} - \frac{4 (x - 1)}{(x + 2) (x - 2)} = 0$

Step 2.4

Write each expression with a common denominator of $(x + 2) x$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

Multiply $\frac{3}{x + 2}$ by $\frac{x}{x}$ .

$\frac{3 x}{(x + 2) x} + \frac{2}{x} \cdot \frac{x + 2}{x + 2} - \frac{4 (x - 1)}{(x + 2) (x - 2)} = 0$

Step 2.4.2

Multiply $\frac{2}{x}$ by $\frac{x + 2}{x + 2}$ .

$\frac{3 x}{(x + 2) x} + \frac{2 (x + 2)}{x (x + 2)} - \frac{4 (x - 1)}{(x + 2) (x - 2)} = 0$

Step 2.4.3

Reorder the factors of $(x + 2) x$ .

$\frac{3 x}{x (x + 2)} + \frac{2 (x + 2)}{x (x + 2)} - \frac{4 (x - 1)}{(x + 2) (x - 2)} = 0$

Step 2.5

Combine the numerators over the common denominator.

$\frac{3 x + 2 (x + 2)}{x (x + 2)} - \frac{4 (x - 1)}{(x + 2) (x - 2)} = 0$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Apply the distributive property.

$\frac{3 x + 2 x + 2 \cdot 2}{x (x + 2)} - \frac{4 (x - 1)}{(x + 2) (x - 2)} = 0$

Step 2.6.2

Multiply $2$ by $2$ .

$\frac{3 x + 2 x + 4}{x (x + 2)} - \frac{4 (x - 1)}{(x + 2) (x - 2)} = 0$

Step 2.6.3

Add $3 x$ and $2 x$ .

$\frac{5 x + 4}{x (x + 2)} - \frac{4 (x - 1)}{(x + 2) (x - 2)} = 0$

Step 2.7

To write $\frac{5 x + 4}{x (x + 2)}$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

$\frac{5 x + 4}{x (x + 2)} \cdot \frac{x - 2}{x - 2} - \frac{4 (x - 1)}{(x + 2) (x - 2)} = 0$

Step 2.8

To write $- \frac{4 (x - 1)}{(x + 2) (x - 2)}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{5 x + 4}{x (x + 2)} \cdot \frac{x - 2}{x - 2} - \frac{4 (x - 1)}{(x + 2) (x - 2)} \cdot \frac{x}{x} = 0$

Step 2.9

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

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Step 2.9.1

Multiply $\frac{5 x + 4}{x (x + 2)}$ by $\frac{x - 2}{x - 2}$ .

$\frac{(5 x + 4) (x - 2)}{x (x + 2) (x - 2)} - \frac{4 (x - 1)}{(x + 2) (x - 2)} \cdot \frac{x}{x} = 0$

Step 2.9.2

Multiply $\frac{4 (x - 1)}{(x + 2) (x - 2)}$ by $\frac{x}{x}$ .

$\frac{(5 x + 4) (x - 2)}{x (x + 2) (x - 2)} - \frac{4 (x - 1) x}{(x + 2) (x - 2) x} = 0$

Step 2.9.3

Reorder the factors of $(x + 2) (x - 2) x$ .

$\frac{(5 x + 4) (x - 2)}{x (x + 2) (x - 2)} - \frac{4 (x - 1) x}{x (x + 2) (x - 2)} = 0$

Step 2.10

Combine the numerators over the common denominator.

$\frac{(5 x + 4) (x - 2) - 4 (x - 1) x}{x (x + 2) (x - 2)} = 0$

Step 2.11

Simplify the numerator.

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Step 2.11.1

Expand $(5 x + 4) (x - 2)$ using the FOIL Method.

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Step 2.11.1.1

Apply the distributive property.

$\frac{5 x (x - 2) + 4 (x - 2) - 4 (x - 1) x}{x (x + 2) (x - 2)} = 0$

Step 2.11.1.2

Apply the distributive property.

$\frac{5 x \cdot x + 5 x \cdot - 2 + 4 (x - 2) - 4 (x - 1) x}{x (x + 2) (x - 2)} = 0$

Step 2.11.1.3

Apply the distributive property.

$\frac{5 x \cdot x + 5 x \cdot - 2 + 4 x + 4 \cdot - 2 - 4 (x - 1) x}{x (x + 2) (x - 2)} = 0$

Step 2.11.2

Simplify and combine like terms.

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Step 2.11.2.1

Simplify each term.

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Step 2.11.2.1.1

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

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Step 2.11.2.1.1.1

Move $x$ .

$\frac{5 (x \cdot x) + 5 x \cdot - 2 + 4 x + 4 \cdot - 2 - 4 (x - 1) x}{x (x + 2) (x - 2)} = 0$

Step 2.11.2.1.1.2

Multiply $x$ by $x$ .

$\frac{5 x^{2} + 5 x \cdot - 2 + 4 x + 4 \cdot - 2 - 4 (x - 1) x}{x (x + 2) (x - 2)} = 0$

Step 2.11.2.1.2

Multiply $- 2$ by $5$ .

$\frac{5 x^{2} - 10 x + 4 x + 4 \cdot - 2 - 4 (x - 1) x}{x (x + 2) (x - 2)} = 0$

Step 2.11.2.1.3

Multiply $4$ by $- 2$ .

$\frac{5 x^{2} - 10 x + 4 x - 8 - 4 (x - 1) x}{x (x + 2) (x - 2)} = 0$

Step 2.11.2.2

Add $- 10 x$ and $4 x$ .

$\frac{5 x^{2} - 6 x - 8 - 4 (x - 1) x}{x (x + 2) (x - 2)} = 0$

Step 2.11.3

Apply the distributive property.

$\frac{5 x^{2} - 6 x - 8 + (- 4 x - 4 \cdot - 1) x}{x (x + 2) (x - 2)} = 0$

Step 2.11.4

Multiply $- 4$ by $- 1$ .

$\frac{5 x^{2} - 6 x - 8 + (- 4 x + 4) x}{x (x + 2) (x - 2)} = 0$

Step 2.11.5

Apply the distributive property.

$\frac{5 x^{2} - 6 x - 8 - 4 x \cdot x + 4 x}{x (x + 2) (x - 2)} = 0$

Step 2.11.6

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

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Step 2.11.6.1

Move $x$ .

$\frac{5 x^{2} - 6 x - 8 - 4 (x \cdot x) + 4 x}{x (x + 2) (x - 2)} = 0$

Step 2.11.6.2

Multiply $x$ by $x$ .

$\frac{5 x^{2} - 6 x - 8 - 4 x^{2} + 4 x}{x (x + 2) (x - 2)} = 0$

Step 2.11.7

Subtract $4 x^{2}$ from $5 x^{2}$ .

$\frac{x^{2} - 6 x - 8 + 4 x}{x (x + 2) (x - 2)} = 0$

Step 2.11.8

Add $- 6 x$ and $4 x$ .

$\frac{x^{2} - 2 x - 8}{x (x + 2) (x - 2)} = 0$

Step 2.11.9

Factor $x^{2} - 2 x - 8$ using the AC method.

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Step 2.11.9.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 8$ and whose sum is $- 2$ .

$- 4, 2$

Step 2.11.9.2

Write the factored form using these integers.

$\frac{(x - 4) (x + 2)}{x (x + 2) (x - 2)} = 0$

Step 2.12

Cancel the common factor of $x + 2$ .

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Step 2.12.1

Cancel the common factor.

$\frac{(x - 4) (x + 2)}{x (x + 2) (x - 2)} = 0$

Step 2.12.2

Rewrite the expression.

$\frac{x - 4}{x (x - 2)} = 0$

Step 3

Set the denominator in $\frac{x - 4}{x (x - 2)}$ equal to $0$ to find where the expression is undefined.

$x (x - 2) = 0$

Step 4

Solve for $x$ .

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Step 4.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x - 2 = 0$

Step 4.2

Set $x$ equal to $0$ .

$x = 0$

Step 4.3

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 4.3.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 4.3.2

Add $2$ to both sides of the equation.

$x = 2$

Step 4.4

The final solution is all the values that make $x (x - 2) = 0$ true.

$x = 0, 2$

Step 5

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x = 0, x = 2$

Step 6