Algebra Examples

$\frac{x}{x + 6} = \frac{72}{x^{2} - 36} + 4$

Step 1

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

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Step 1.1

Subtract $\frac{72}{x^{2} - 36}$ from both sides of the equation.

$\frac{x}{x + 6} - \frac{72}{x^{2} - 36} = 4$

Step 1.2

Subtract $4$ from both sides of the equation.

$\frac{x}{x + 6} - \frac{72}{x^{2} - 36} - 4 = 0$

Step 2

Simplify $\frac{x}{x + 6} - \frac{72}{x^{2} - 36} - 4$ .

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Step 2.1

Simplify the denominator.

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Step 2.1.1

Rewrite $36$ as $6^{2}$ .

$\frac{x}{x + 6} - \frac{72}{x^{2} - 6^{2}} - 4 = 0$

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 $b = 6$ .

$\frac{x}{x + 6} - \frac{72}{(x + 6) (x - 6)} - 4 = 0$

Step 2.2

To write $\frac{x}{x + 6}$ as a fraction with a common denominator, multiply by $\frac{x - 6}{x - 6}$ .

$\frac{x}{x + 6} \cdot \frac{x - 6}{x - 6} - \frac{72}{(x + 6) (x - 6)} - 4 = 0$

Step 2.3

Multiply $\frac{x}{x + 6}$ by $\frac{x - 6}{x - 6}$ .

$\frac{x (x - 6)}{(x + 6) (x - 6)} - \frac{72}{(x + 6) (x - 6)} - 4 = 0$

Step 2.4

Combine the numerators over the common denominator.

$\frac{x (x - 6) - 72}{(x + 6) (x - 6)} - 4 = 0$

Step 2.5

Simplify each term.

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 6 - 72}{(x + 6) (x - 6)} - 4 = 0$

Step 2.5.1.2

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot - 6 - 72}{(x + 6) (x - 6)} - 4 = 0$

Step 2.5.1.3

Move $- 6$ to the left of $x$ .

$\frac{x^{2} - 6 \cdot x - 72}{(x + 6) (x - 6)} - 4 = 0$

Step 2.5.1.4

Factor $x^{2} - 6 x - 72$ using the AC method.

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Step 2.5.1.4.1

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 $- 72$ and whose sum is $- 6$ .

$- 12, 6$

Step 2.5.1.4.2

Write the factored form using these integers.

$\frac{(x - 12) (x + 6)}{(x + 6) (x - 6)} - 4 = 0$

Step 2.5.2

Cancel the common factor of $x + 6$ .

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Step 2.5.2.1

Cancel the common factor.

$\frac{(x - 12) (x + 6)}{(x + 6) (x - 6)} - 4 = 0$

Step 2.5.2.2

Rewrite the expression.

$\frac{x - 12}{x - 6} - 4 = 0$

Step 2.6

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{x - 6}{x - 6}$ .

$\frac{x - 12}{x - 6} - 4 \cdot \frac{x - 6}{x - 6} = 0$

Step 2.7

Combine $- 4$ and $\frac{x - 6}{x - 6}$ .

$\frac{x - 12}{x - 6} + \frac{- 4 (x - 6)}{x - 6} = 0$

Step 2.8

Combine the numerators over the common denominator.

$\frac{x - 12 - 4 (x - 6)}{x - 6} = 0$

Step 2.9

Simplify the numerator.

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Step 2.9.1

Apply the distributive property.

$\frac{x - 12 - 4 x - 4 \cdot - 6}{x - 6} = 0$

Step 2.9.2

Multiply $- 4$ by $- 6$ .

$\frac{x - 12 - 4 x + 24}{x - 6} = 0$

Step 2.9.3

Subtract $4 x$ from $x$ .

$\frac{- 3 x - 12 + 24}{x - 6} = 0$

Step 2.9.4

Add $- 12$ and $24$ .

$\frac{- 3 x + 12}{x - 6} = 0$

Step 2.9.5

Factor $3$ out of $- 3 x + 12$ .

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Step 2.9.5.1

Factor $3$ out of $- 3 x$ .

$\frac{3 (- x) + 12}{x - 6} = 0$

Step 2.9.5.2

Factor $3$ out of $12$ .

$\frac{3 (- x) + 3 (4)}{x - 6} = 0$

Step 2.9.5.3

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

$\frac{3 (- x + 4)}{x - 6} = 0$

Step 2.10

Factor $- 1$ out of $- x$ .

$\frac{3 (- (x) + 4)}{x - 6} = 0$

Step 2.11

Rewrite $4$ as $- 1 (- 4)$ .

$\frac{3 (- (x) - 1 (- 4))}{x - 6} = 0$

Step 2.12

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

$\frac{3 (- (x - 4))}{x - 6} = 0$

Step 2.13

Rewrite $- (x - 4)$ as $- 1 (x - 4)$ .

$\frac{3 (- 1 (x - 4))}{x - 6} = 0$

Step 2.14

Move the negative in front of the fraction.

$- \frac{3 (x - 4)}{x - 6} = 0$

Step 3

Set the numerator equal to zero.

$3 (x - 4) = 0$

Step 4

Solve the equation for $x$ .

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Step 4.1

Divide each term in $3 (x - 4) = 0$ by $3$ and simplify.

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Step 4.1.1

Divide each term in $3 (x - 4) = 0$ by $3$ .

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

Step 4.1.2

Simplify the left side.

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Step 4.1.2.1

Cancel the common factor of $3$ .

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Step 4.1.2.1.1

Cancel the common factor.

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

Step 4.1.2.1.2

Divide $x - 4$ by $1$ .

$x - 4 = \frac{0}{3}$

Step 4.1.3

Simplify the right side.

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Step 4.1.3.1

Divide $0$ by $3$ .

$x - 4 = 0$

Step 4.2

Add $4$ to both sides of the equation.

$x = 4$