Solve by Factoring 2/(x+5)+1/(x-5)=16/(x^2-25)

$\frac{2}{x + 5} + \frac{1}{x - 5} = \frac{16}{x^{2} - 25}$

Step 1

Subtract $\frac{16}{x^{2} - 25}$ from both sides of the equation.

$\frac{2}{x + 5} + \frac{1}{x - 5} - \frac{16}{x^{2} - 25} = 0$

Step 2

Simplify $\frac{2}{x + 5} + \frac{1}{x - 5} - \frac{16}{x^{2} - 25}$ .

Simplify the denominator.

Rewrite $25$ as $5^{2}$ .

$\frac{2}{x + 5} + \frac{1}{x - 5} - \frac{16}{x^{2} - 5^{2}} = 0$

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 $b = 5$ .

$\frac{2}{x + 5} + \frac{1}{x - 5} - \frac{16}{(x + 5) (x - 5)} = 0$

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

$\frac{2}{x + 5} \cdot \frac{x - 5}{x - 5} + \frac{1}{x - 5} - \frac{16}{(x + 5) (x - 5)} = 0$

To write $\frac{1}{x - 5}$ as a fraction with a common denominator, multiply by $\frac{x + 5}{x + 5}$ .

$\frac{2}{x + 5} \cdot \frac{x - 5}{x - 5} + \frac{1}{x - 5} \cdot \frac{x + 5}{x + 5} - \frac{16}{(x + 5) (x - 5)} = 0$

Write each expression with a common denominator of $(x + 5) (x - 5)$ , by multiplying each by an appropriate factor of $1$ .

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

$\frac{2 (x - 5)}{(x + 5) (x - 5)} + \frac{1}{x - 5} \cdot \frac{x + 5}{x + 5} - \frac{16}{(x + 5) (x - 5)} = 0$

Multiply $\frac{1}{x - 5}$ by $\frac{x + 5}{x + 5}$ .

$\frac{2 (x - 5)}{(x + 5) (x - 5)} + \frac{x + 5}{(x - 5) (x + 5)} - \frac{16}{(x + 5) (x - 5)} = 0$

Reorder the factors of $(x - 5) (x + 5)$ .

$\frac{2 (x - 5)}{(x + 5) (x - 5)} + \frac{x + 5}{(x + 5) (x - 5)} - \frac{16}{(x + 5) (x - 5)} = 0$

Combine the numerators over the common denominator.

$\frac{2 (x - 5) + x + 5}{(x + 5) (x - 5)} - \frac{16}{(x + 5) (x - 5)} = 0$

Combine the numerators over the common denominator.

$\frac{2 (x - 5) + x + 5 - 16}{(x + 5) (x - 5)} = 0$

Simplify each term.

Apply the distributive property.

$\frac{2 x + 2 \cdot - 5 + x + 5 - 16}{(x + 5) (x - 5)} = 0$

Multiply $2$ by $- 5$ .

$\frac{2 x - 10 + x + 5 - 16}{(x + 5) (x - 5)} = 0$

Add $2 x$ and $x$ .

$\frac{3 x - 10 + 5 - 16}{(x + 5) (x - 5)} = 0$

Add $- 10$ and $5$ .

$\frac{3 x - 5 - 16}{(x + 5) (x - 5)} = 0$

Subtract $16$ from $- 5$ .

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

Factor $3$ out of $3 x - 21$ .

Factor $3$ out of $3 x$ .

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

Factor $3$ out of $- 21$ .

$\frac{3 x + 3 \cdot - 7}{(x + 5) (x - 5)} = 0$

Factor $3$ out of $3 x + 3 \cdot - 7$ .

$\frac{3 (x - 7)}{(x + 5) (x - 5)} = 0$

Step 3

Set the numerator equal to zero.

$3 (x - 7) = 0$

Step 4

Solve the equation for $x$ .

Divide each term in $3 (x - 7) = 0$ by $3$ and simplify.

Divide each term in $3 (x - 7) = 0$ by $3$ .

$\frac{3 (x - 7)}{3} = \frac{0}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 (x - 7)}{3} = \frac{0}{3}$

Divide $x - 7$ by $1$ .

$x - 7 = \frac{0}{3}$

Simplify the right side.

Divide $0$ by $3$ .

$x - 7 = 0$

Add $7$ to both sides of the equation.

$x = 7$

Solve by Factoring 2/(x+5)+1/(x-5)=16/(x^2-25)

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