Algebra Examples

$\frac{3 x}{x + 1} = \frac{12}{x^{2} - 1} + 2$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $\frac{12}{x^{2} - 1}$ from both sides of the equation.

$\frac{3 x}{x + 1} - \frac{12}{x^{2} - 1} = 2$

Step 1.2

Subtract $2$ from both sides of the equation.

$\frac{3 x}{x + 1} - \frac{12}{x^{2} - 1} - 2 = 0$

Step 2

Simplify $\frac{3 x}{x + 1} - \frac{12}{x^{2} - 1} - 2$ .

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Step 2.1

Simplify the denominator.

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Step 2.1.1

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

$\frac{3 x}{x + 1} - \frac{12}{x^{2} - 1^{2}} - 2 = 0$

Step 2.1.2

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 $b = 1$ .

$\frac{3 x}{x + 1} - \frac{12}{(x + 1) (x - 1)} - 2 = 0$

Step 2.2

To write $\frac{3 x}{x + 1}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$\frac{3 x}{x + 1} \cdot \frac{x - 1}{x - 1} - \frac{12}{(x + 1) (x - 1)} - 2 = 0$

Step 2.3

Multiply $\frac{3 x}{x + 1}$ by $\frac{x - 1}{x - 1}$ .

$\frac{3 x (x - 1)}{(x + 1) (x - 1)} - \frac{12}{(x + 1) (x - 1)} - 2 = 0$

Step 2.4

Combine the numerators over the common denominator.

$\frac{3 x (x - 1) - 12}{(x + 1) (x - 1)} - 2 = 0$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Factor $3$ out of $3 x (x - 1) - 12$ .

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Step 2.5.1.1

Factor $3$ out of $3 x (x - 1)$ .

$\frac{3 (x (x - 1)) - 12}{(x + 1) (x - 1)} - 2 = 0$

Step 2.5.1.2

Factor $3$ out of $- 12$ .

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

Step 2.5.1.3

Factor $3$ out of $3 (x (x - 1)) + 3 \cdot - 4$ .

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

Step 2.5.2

Apply the distributive property.

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

Step 2.5.3

Multiply $x$ by $x$ .

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

Step 2.5.4

Move $- 1$ to the left of $x$ .

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

Step 2.5.5

Rewrite $- 1 x$ as $- x$ .

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

Step 2.6

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

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

Step 2.7

Combine $- 2$ and $\frac{(x + 1) (x - 1)}{(x + 1) (x - 1)}$ .

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

Step 2.8

Combine the numerators over the common denominator.

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

Step 2.9

Simplify the numerator.

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Step 2.9.1

Apply the distributive property.

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

Step 2.9.2

Simplify.

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Step 2.9.2.1

Multiply $- 1$ by $3$ .

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

Step 2.9.2.2

Multiply $3$ by $- 4$ .

$\frac{3 x^{2} - 3 x - 12 - 2 (x + 1) (x - 1)}{(x + 1) (x - 1)} = 0$

Step 2.9.3

Apply the distributive property.

$\frac{3 x^{2} - 3 x - 12 + (- 2 x - 2 \cdot 1) (x - 1)}{(x + 1) (x - 1)} = 0$

Step 2.9.4

Multiply $- 2$ by $1$ .

$\frac{3 x^{2} - 3 x - 12 + (- 2 x - 2) (x - 1)}{(x + 1) (x - 1)} = 0$

Step 2.9.5

Expand $(- 2 x - 2) (x - 1)$ using the FOIL Method.

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Step 2.9.5.1

Apply the distributive property.

$\frac{3 x^{2} - 3 x - 12 - 2 x (x - 1) - 2 (x - 1)}{(x + 1) (x - 1)} = 0$

Step 2.9.5.2

Apply the distributive property.

$\frac{3 x^{2} - 3 x - 12 - 2 x \cdot x - 2 x \cdot - 1 - 2 (x - 1)}{(x + 1) (x - 1)} = 0$

Step 2.9.5.3

Apply the distributive property.

$\frac{3 x^{2} - 3 x - 12 - 2 x \cdot x - 2 x \cdot - 1 - 2 x - 2 \cdot - 1}{(x + 1) (x - 1)} = 0$

Step 2.9.6

Simplify and combine like terms.

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Step 2.9.6.1

Simplify each term.

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Step 2.9.6.1.1

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

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Step 2.9.6.1.1.1

Move $x$ .

$\frac{3 x^{2} - 3 x - 12 - 2 (x \cdot x) - 2 x \cdot - 1 - 2 x - 2 \cdot - 1}{(x + 1) (x - 1)} = 0$

Step 2.9.6.1.1.2

Multiply $x$ by $x$ .

$\frac{3 x^{2} - 3 x - 12 - 2 x^{2} - 2 x \cdot - 1 - 2 x - 2 \cdot - 1}{(x + 1) (x - 1)} = 0$

Step 2.9.6.1.2

Multiply $- 1$ by $- 2$ .

$\frac{3 x^{2} - 3 x - 12 - 2 x^{2} + 2 x - 2 x - 2 \cdot - 1}{(x + 1) (x - 1)} = 0$

Step 2.9.6.1.3

Multiply $- 2$ by $- 1$ .

$\frac{3 x^{2} - 3 x - 12 - 2 x^{2} + 2 x - 2 x + 2}{(x + 1) (x - 1)} = 0$

Step 2.9.6.2

Subtract $2 x$ from $2 x$ .

$\frac{3 x^{2} - 3 x - 12 - 2 x^{2} + 0 + 2}{(x + 1) (x - 1)} = 0$

Step 2.9.6.3

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

$\frac{3 x^{2} - 3 x - 12 - 2 x^{2} + 2}{(x + 1) (x - 1)} = 0$

Step 2.9.7

Subtract $2 x^{2}$ from $3 x^{2}$ .

$\frac{x^{2} - 3 x - 12 + 2}{(x + 1) (x - 1)} = 0$

Step 2.9.8

Add $- 12$ and $2$ .

$\frac{x^{2} - 3 x - 10}{(x + 1) (x - 1)} = 0$

Step 2.9.9

Factor $x^{2} - 3 x - 10$ using the AC method.

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Step 2.9.9.1

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 $- 10$ and whose sum is $- 3$ .

$- 5, 2$

Step 2.9.9.2

Write the factored form using these integers.

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

Step 3

Set the numerator equal to zero.

$(x - 5) (x + 2) = 0$

Step 4

Solve the equation 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 $0$ .

$x - 5 = 0$

$x + 2 = 0$

Step 4.2

Set $x - 5$ equal to $0$ and solve for $x$ .

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Step 4.2.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 4.2.2

Add $5$ to both sides of the equation.

$x = 5$

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

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 4.4

The final solution is all the values that make $(x - 5) (x + 2) = 0$ true.

$x = 5, - 2$