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

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

Step 1

Subtract $1$ from both sides of the equation.

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

Step 2

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

Simplify the denominator.

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

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

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 $b = 5$ .

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

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

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

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

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

Combine the numerators over the common denominator.

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

Add $- 5$ and $10$ .

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

Cancel the common factor of $x + 5$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{1}{x - 5} - 1 = 0$

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

$\frac{1}{x - 5} - 1 \cdot \frac{x - 5}{x - 5} = 0$

Combine $- 1$ and $\frac{x - 5}{x - 5}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

$\frac{1 - x - - 5}{x - 5} = 0$

Multiply $- 1$ by $- 5$ .

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

Add $1$ and $5$ .

$\frac{- x + 6}{x - 5} = 0$

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

$\frac{- (x) + 6}{x - 5} = 0$

Rewrite $6$ as $- 1 (- 6)$ .

$\frac{- (x) - 1 (- 6)}{x - 5} = 0$

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

$\frac{- (x - 6)}{x - 5} = 0$

Rewrite $- (x - 6)$ as $- 1 (x - 6)$ .

$\frac{- 1 (x - 6)}{x - 5} = 0$

Move the negative in front of the fraction.

$- \frac{x - 6}{x - 5} = 0$

Step 3

Set the numerator equal to zero.

$x - 6 = 0$

Step 4

Add $6$ to both sides of the equation.

$x = 6$

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

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