Solve by Factoring 1/(x-5)+1/(x+5)=10/(x^2-25)

$\frac{1}{x - 5} + \frac{1}{x + 5} = \frac{10}{x^{2} - 25}$

Step 1

Subtract $\frac{10}{x^{2} - 25}$ from both sides of the equation.

$\frac{1}{x - 5} + \frac{1}{x + 5} - \frac{10}{x^{2} - 25} = 0$

Step 2

Simplify $\frac{1}{x - 5} + \frac{1}{x + 5} - \frac{10}{x^{2} - 25}$ .

Simplify the denominator.

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

$\frac{1}{x - 5} + \frac{1}{x + 5} - \frac{10}{x^{2} - 5^{2}} = 0$

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 $b = 5$ .

$\frac{1}{x - 5} + \frac{1}{x + 5} - \frac{10}{(x + 5) (x - 5)} = 0$

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

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

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

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Combine the opposite terms in $\frac{x + 5 + x - 5}{(x + 5) (x - 5)} - \frac{10}{(x + 5) (x - 5)}$ .

Subtract $5$ from $5$ .

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

Add $x$ and $0$ .

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

Combine the numerators over the common denominator.

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

Add $x$ and $x$ .

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

Factor $2$ out of $2 x - 10$ .

Factor $2$ out of $2 x$ .

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

Factor $2$ out of $- 10$ .

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

Factor $2$ out of $2 x + 2 \cdot - 5$ .

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

Cancel the common factor of $x - 5$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{2}{x + 5} = 0$

Step 3

Set the numerator equal to zero.

$2 = 0$

Step 4

Since $2 \neq 0$ , there are no solutions.

No solution

Solve by Factoring 1/(x-5)+1/(x+5)=10/(x^2-25)

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