Solve by Factoring 36/(x^2-9)-3=(2x)/(x-3)

$\frac{36}{x^{2} - 9} - 3 = \frac{2 x}{x - 3}$

Step 1

Subtract $\frac{2 x}{x - 3}$ from both sides of the equation.

$\frac{36}{x^{2} - 9} - 3 - \frac{2 x}{x - 3} = 0$

Step 2

Simplify $\frac{36}{x^{2} - 9} - 3 - \frac{2 x}{x - 3}$ .

Simplify the denominator.

Rewrite $9$ as $3^{2}$ .

$\frac{36}{x^{2} - 3^{2}} - 3 - \frac{2 x}{x - 3} = 0$

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 $b = 3$ .

$\frac{36}{(x + 3) (x - 3)} - 3 - \frac{2 x}{x - 3} = 0$

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

$\frac{36}{(x + 3) (x - 3)} - \frac{2 x}{x - 3} \cdot \frac{x + 3}{x + 3} - 3 = 0$

Write each expression with a common denominator of $(x + 3) (x - 3)$ , by multiplying each by an appropriate factor of $1$ .

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

$\frac{36}{(x + 3) (x - 3)} - \frac{2 x (x + 3)}{(x - 3) (x + 3)} - 3 = 0$

Reorder the factors of $(x - 3) (x + 3)$ .

$\frac{36}{(x + 3) (x - 3)} - \frac{2 x (x + 3)}{(x + 3) (x - 3)} - 3 = 0$

Combine the numerators over the common denominator.

$\frac{36 - 2 x (x + 3)}{(x + 3) (x - 3)} - 3 = 0$

Simplify each term.

Simplify the numerator.

Factor $2$ out of $36 - 2 x (x + 3)$ .

Factor $2$ out of $36$ .

$\frac{2 (18) - 2 x (x + 3)}{(x + 3) (x - 3)} - 3 = 0$

Factor $2$ out of $- 2 x (x + 3)$ .

$\frac{2 (18) + 2 (- x (x + 3))}{(x + 3) (x - 3)} - 3 = 0$

Factor $2$ out of $2 (18) + 2 (- x (x + 3))$ .

$\frac{2 (18 - x (x + 3))}{(x + 3) (x - 3)} - 3 = 0$

Apply the distributive property.

$\frac{2 (18 - x \cdot x - x \cdot 3)}{(x + 3) (x - 3)} - 3 = 0$

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

Move $x$ .

$\frac{2 (18 - (x \cdot x) - x \cdot 3)}{(x + 3) (x - 3)} - 3 = 0$

Multiply $x$ by $x$ .

$\frac{2 (18 - x^{2} - x \cdot 3)}{(x + 3) (x - 3)} - 3 = 0$

Multiply $3$ by $- 1$ .

$\frac{2 (18 - x^{2} - 3 x)}{(x + 3) (x - 3)} - 3 = 0$

Reorder terms.

$\frac{2 (- x^{2} - 3 x + 18)}{(x + 3) (x - 3)} - 3 = 0$

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 18 = - 18$ and whose sum is $b = - 3$ .

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

$\frac{2 (- x^{2} - 3 (x) + 18)}{(x + 3) (x - 3)} - 3 = 0$

Rewrite $- 3$ as $3$ plus $- 6$

$\frac{2 (- x^{2} + (3 - 6) x + 18)}{(x + 3) (x - 3)} - 3 = 0$

Apply the distributive property.

$\frac{2 (- x^{2} + 3 x - 6 x + 18)}{(x + 3) (x - 3)} - 3 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{2 ((- x^{2} + 3 x) - 6 x + 18)}{(x + 3) (x - 3)} - 3 = 0$

Factor out the greatest common factor (GCF) from each group.

$\frac{2 (x (- x + 3) + 6 (- x + 3))}{(x + 3) (x - 3)} - 3 = 0$

Factor the polynomial by factoring out the greatest common factor, $- x + 3$ .

$\frac{2 ((- x + 3) (x + 6))}{(x + 3) (x - 3)} - 3 = 0$

$\frac{2 (- x + 3) (x + 6)}{(x + 3) (x - 3)} - 3 = 0$

Cancel the common factor of $- x + 3$ and $x - 3$ .

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

$\frac{2 (- (x) + 3) (x + 6)}{(x + 3) (x - 3)} - 3 = 0$

Rewrite $3$ as $- 1 (- 3)$ .

$\frac{2 (- (x) - 1 (- 3)) (x + 6)}{(x + 3) (x - 3)} - 3 = 0$

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

$\frac{2 (- (x - 3)) (x + 6)}{(x + 3) (x - 3)} - 3 = 0$

Rewrite $- (x - 3)$ as $- 1 (x - 3)$ .

$\frac{2 (- 1 (x - 3)) (x + 6)}{(x + 3) (x - 3)} - 3 = 0$

Cancel the common factor.

$\frac{2 (- 1 (x - 3)) (x + 6)}{(x + 3) (x - 3)} - 3 = 0$

Rewrite the expression.

$\frac{2 \cdot (- 1) (x + 6)}{x + 3} - 3 = 0$

Multiply $2$ by $- 1$ .

$\frac{- 2 (x + 6)}{x + 3} - 3 = 0$

Move the negative in front of the fraction.

$- \frac{2 (x + 6)}{x + 3} - 3 = 0$

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

$- \frac{2 (x + 6)}{x + 3} - 3 \cdot \frac{x + 3}{x + 3} = 0$

Combine $- 3$ and $\frac{x + 3}{x + 3}$ .

$- \frac{2 (x + 6)}{x + 3} + \frac{- 3 (x + 3)}{x + 3} = 0$

Combine the numerators over the common denominator.

$\frac{- 2 (x + 6) - 3 (x + 3)}{x + 3} = 0$

Simplify the numerator.

Factor $- 1$ out of $- 2 (x + 6) - 3 (x + 3)$ .

Factor $- 1$ out of $- 2 (x + 6)$ .

$\frac{- (2 (x + 6)) - 3 (x + 3)}{x + 3} = 0$

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

$\frac{- (2 (x + 6)) - (3 (x + 3))}{x + 3} = 0$

Factor $- 1$ out of $- (2 (x + 6)) - (3 (x + 3))$ .

$\frac{- (2 (x + 6) + 3 (x + 3))}{x + 3} = 0$

Apply the distributive property.

$\frac{- (2 x + 2 \cdot 6 + 3 (x + 3))}{x + 3} = 0$

Multiply $2$ by $6$ .

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

Apply the distributive property.

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

Multiply $3$ by $3$ .

$\frac{- (2 x + 12 + 3 x + 9)}{x + 3} = 0$

Add $2 x$ and $3 x$ .

$\frac{- (5 x + 12 + 9)}{x + 3} = 0$

Add $12$ and $9$ .

$\frac{- (5 x + 21)}{x + 3} = 0$

Move the negative in front of the fraction.

$- \frac{5 x + 21}{x + 3} = 0$

Step 3

Set the numerator equal to zero.

$5 x + 21 = 0$

Step 4

Solve the equation for $x$ .

Subtract $21$ from both sides of the equation.

$5 x = - 21$

Divide each term in $5 x = - 21$ by $5$ and simplify.

Divide each term in $5 x = - 21$ by $5$ .

$\frac{5 x}{5} = \frac{- 21}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 21}{5}$

Divide $x$ by $1$ .

$x = \frac{- 21}{5}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{21}{5}$

Solve by Factoring 36/(x^2-9)-3=(2x)/(x-3)

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