Solve by Factoring 1/x+1/(x+2)=1/5

$\frac{1}{x} + \frac{1}{x + 2} = \frac{1}{5}$

Step 1

Subtract $\frac{1}{5}$ from both sides of the equation.

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

Step 2

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

Find the common denominator.

Multiply $\frac{1}{x}$ by $\frac{(x + 2) \cdot 5}{(x + 2) \cdot 5}$ .

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

Multiply $\frac{1}{x}$ by $\frac{(x + 2) \cdot 5}{(x + 2) \cdot 5}$ .

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

Multiply $\frac{1}{x + 2}$ by $\frac{x \cdot 5}{x \cdot 5}$ .

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

Multiply $\frac{1}{x + 2}$ by $\frac{x \cdot 5}{x \cdot 5}$ .

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

Multiply $\frac{1}{5}$ by $\frac{x (x + 2)}{x (x + 2)}$ .

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

Multiply $\frac{1}{5}$ by $\frac{x (x + 2)}{x (x + 2)}$ .

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

Reorder the factors of $x ((x + 2) \cdot 5)$ .

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

Reorder the factors of $(x + 2) (x \cdot 5)$ .

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

Reorder the factors of $5 (x (x + 2))$ .

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

Combine the numerators over the common denominator.

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

Simplify each term.

Apply the distributive property.

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

Move $5$ to the left of $x$ .

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

Multiply $2$ by $5$ .

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

Move $5$ to the left of $x$ .

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

Apply the distributive property.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $2$ by $- 1$ .

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

Add $5 x$ and $5 x$ .

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

Subtract $2 x$ from $10 x$ .

$\frac{8 x + 10 - x^{2}}{5 x (x + 2)} = 0$

Reorder terms.

$\frac{- x^{2} + 8 x + 10}{5 x (x + 2)} = 0$

Factor $- 1$ out of $- x^{2}$ .

$\frac{- (x^{2}) + 8 x + 10}{5 x (x + 2)} = 0$

Factor $- 1$ out of $8 x$ .

$\frac{- (x^{2}) - (- 8 x) + 10}{5 x (x + 2)} = 0$

Factor $- 1$ out of $- (x^{2}) - (- 8 x)$ .

$\frac{- (x^{2} - 8 x) + 10}{5 x (x + 2)} = 0$

Rewrite $10$ as $- 1 (- 10)$ .

$\frac{- (x^{2} - 8 x) - 1 (- 10)}{5 x (x + 2)} = 0$

Factor $- 1$ out of $- (x^{2} - 8 x) - 1 (- 10)$ .

$\frac{- (x^{2} - 8 x - 10)}{5 x (x + 2)} = 0$

Rewrite $- (x^{2} - 8 x - 10)$ as $- 1 (x^{2} - 8 x - 10)$ .

$\frac{- 1 (x^{2} - 8 x - 10)}{5 x (x + 2)} = 0$

Move the negative in front of the fraction.

$- \frac{x^{2} - 8 x - 10}{5 x (x + 2)} = 0$

Step 3

Set the numerator equal to zero.

$x^{2} - 8 x - 10 = 0$

Step 4

Solve the equation for $x$ .

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 1$ , $b = - 8$ , and $c = - 10$ into the quadratic formula and solve for $x$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (1 \cdot - 10)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot - 10}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 10$ .

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot - 10}}{2 \cdot 1}$

Multiply $- 4$ by $- 10$ .

$x = \frac{8 \pm \sqrt{64 + 40}}{2 \cdot 1}$

Add $64$ and $40$ .

$x = \frac{8 \pm \sqrt{104}}{2 \cdot 1}$

Rewrite $104$ as $2^{2} \cdot 26$ .

Factor $4$ out of $104$ .

$x = \frac{8 \pm \sqrt{4 (26)}}{2 \cdot 1}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{8 \pm \sqrt{2^{2} \cdot 26}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{8 \pm 2 \sqrt{26}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{8 \pm 2 \sqrt{26}}{2}$

Simplify $\frac{8 \pm 2 \sqrt{26}}{2}$ .

$x = 4 \pm \sqrt{26}$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot - 10}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 10$ .

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot - 10}}{2 \cdot 1}$

Multiply $- 4$ by $- 10$ .

$x = \frac{8 \pm \sqrt{64 + 40}}{2 \cdot 1}$

Add $64$ and $40$ .

$x = \frac{8 \pm \sqrt{104}}{2 \cdot 1}$

Rewrite $104$ as $2^{2} \cdot 26$ .

Factor $4$ out of $104$ .

$x = \frac{8 \pm \sqrt{4 (26)}}{2 \cdot 1}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{8 \pm \sqrt{2^{2} \cdot 26}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{8 \pm 2 \sqrt{26}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{8 \pm 2 \sqrt{26}}{2}$

Simplify $\frac{8 \pm 2 \sqrt{26}}{2}$ .

$x = 4 \pm \sqrt{26}$

Change the $\pm$ to $+$ .

$x = 4 + \sqrt{26}$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot - 10}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 10$ .

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot - 10}}{2 \cdot 1}$

Multiply $- 4$ by $- 10$ .

$x = \frac{8 \pm \sqrt{64 + 40}}{2 \cdot 1}$

Add $64$ and $40$ .

$x = \frac{8 \pm \sqrt{104}}{2 \cdot 1}$

Rewrite $104$ as $2^{2} \cdot 26$ .

Factor $4$ out of $104$ .

$x = \frac{8 \pm \sqrt{4 (26)}}{2 \cdot 1}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{8 \pm \sqrt{2^{2} \cdot 26}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{8 \pm 2 \sqrt{26}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{8 \pm 2 \sqrt{26}}{2}$

Simplify $\frac{8 \pm 2 \sqrt{26}}{2}$ .

$x = 4 \pm \sqrt{26}$

Change the $\pm$ to $-$ .

$x = 4 - \sqrt{26}$

The final answer is the combination of both solutions.

$x = 4 + \sqrt{26}, 4 - \sqrt{26}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = 4 + \sqrt{26}, 4 - \sqrt{26}$

Decimal Form:

$x = 9.09901951 \dots, - 1.09901951 \dots$

Solve by Factoring 1/x+1/(x+2)=1/5

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