Solve by Factoring 1/x+1/(x+4)=1/5

$\frac{1}{x} + \frac{1}{x + 4} = \frac{1}{5}$

Step 1

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

$\frac{1}{x} + \frac{1}{x + 4} - \frac{1}{5} = 0$

Step 2

Simplify $\frac{1}{x} + \frac{1}{x + 4} - \frac{1}{5}$ .

Find the common denominator.

Multiply $\frac{1}{x}$ by $\frac{(x + 4) \cdot 5}{(x + 4) \cdot 5}$ .

$\frac{1}{x} \cdot \frac{(x + 4) \cdot 5}{(x + 4) \cdot 5} + \frac{1}{x + 4} - \frac{1}{5} = 0$

Multiply $\frac{1}{x}$ by $\frac{(x + 4) \cdot 5}{(x + 4) \cdot 5}$ .

$\frac{(x + 4) \cdot 5}{x ((x + 4) \cdot 5)} + \frac{1}{x + 4} - \frac{1}{5} = 0$

Multiply $\frac{1}{x + 4}$ by $\frac{x \cdot 5}{x \cdot 5}$ .

$\frac{(x + 4) \cdot 5}{x ((x + 4) \cdot 5)} + \frac{1}{x + 4} \cdot \frac{x \cdot 5}{x \cdot 5} - \frac{1}{5} = 0$

Multiply $\frac{1}{x + 4}$ by $\frac{x \cdot 5}{x \cdot 5}$ .

$\frac{(x + 4) \cdot 5}{x ((x + 4) \cdot 5)} + \frac{x \cdot 5}{(x + 4) (x \cdot 5)} - \frac{1}{5} = 0$

Multiply $\frac{1}{5}$ by $\frac{x (x + 4)}{x (x + 4)}$ .

$\frac{(x + 4) \cdot 5}{x ((x + 4) \cdot 5)} + \frac{x \cdot 5}{(x + 4) (x \cdot 5)} - (\frac{1}{5} \cdot \frac{x (x + 4)}{x (x + 4)}) = 0$

Multiply $\frac{1}{5}$ by $\frac{x (x + 4)}{x (x + 4)}$ .

$\frac{(x + 4) \cdot 5}{x ((x + 4) \cdot 5)} + \frac{x \cdot 5}{(x + 4) (x \cdot 5)} - \frac{x (x + 4)}{5 (x (x + 4))} = 0$

Reorder the factors of $x ((x + 4) \cdot 5)$ .

$\frac{(x + 4) \cdot 5}{5 x (x + 4)} + \frac{x \cdot 5}{(x + 4) (x \cdot 5)} - \frac{x (x + 4)}{5 (x (x + 4))} = 0$

Reorder the factors of $(x + 4) (x \cdot 5)$ .

$\frac{(x + 4) \cdot 5}{5 x (x + 4)} + \frac{x \cdot 5}{5 x (x + 4)} - \frac{x (x + 4)}{5 (x (x + 4))} = 0$

Reorder the factors of $5 (x (x + 4))$ .

$\frac{(x + 4) \cdot 5}{5 x (x + 4)} + \frac{x \cdot 5}{5 x (x + 4)} - \frac{x (x + 4)}{5 x (x + 4)} = 0$

Combine the numerators over the common denominator.

$\frac{(x + 4) \cdot 5 + x \cdot 5 - x (x + 4)}{5 x (x + 4)} = 0$

Simplify each term.

Apply the distributive property.

$\frac{x \cdot 5 + 4 \cdot 5 + x \cdot 5 - x (x + 4)}{5 x (x + 4)} = 0$

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

$\frac{5 \cdot x + 4 \cdot 5 + x \cdot 5 - x (x + 4)}{5 x (x + 4)} = 0$

Multiply $4$ by $5$ .

$\frac{5 \cdot x + 20 + x \cdot 5 - x (x + 4)}{5 x (x + 4)} = 0$

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

$\frac{5 x + 20 + 5 \cdot x - x (x + 4)}{5 x (x + 4)} = 0$

Apply the distributive property.

$\frac{5 x + 20 + 5 x - x \cdot x - x \cdot 4}{5 x (x + 4)} = 0$

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

Move $x$ .

$\frac{5 x + 20 + 5 x - (x \cdot x) - x \cdot 4}{5 x (x + 4)} = 0$

Multiply $x$ by $x$ .

$\frac{5 x + 20 + 5 x - x^{2} - x \cdot 4}{5 x (x + 4)} = 0$

Multiply $4$ by $- 1$ .

$\frac{5 x + 20 + 5 x - x^{2} - 4 x}{5 x (x + 4)} = 0$

Add $5 x$ and $5 x$ .

$\frac{10 x + 20 - x^{2} - 4 x}{5 x (x + 4)} = 0$

Subtract $4 x$ from $10 x$ .

$\frac{6 x + 20 - x^{2}}{5 x (x + 4)} = 0$

Reorder terms.

$\frac{- x^{2} + 6 x + 20}{5 x (x + 4)} = 0$

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

$\frac{- (x^{2}) + 6 x + 20}{5 x (x + 4)} = 0$

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

$\frac{- (x^{2}) - (- 6 x) + 20}{5 x (x + 4)} = 0$

Factor $- 1$ out of $- (x^{2}) - (- 6 x)$ .

$\frac{- (x^{2} - 6 x) + 20}{5 x (x + 4)} = 0$

Rewrite $20$ as $- 1 (- 20)$ .

$\frac{- (x^{2} - 6 x) - 1 (- 20)}{5 x (x + 4)} = 0$

Factor $- 1$ out of $- (x^{2} - 6 x) - 1 (- 20)$ .

$\frac{- (x^{2} - 6 x - 20)}{5 x (x + 4)} = 0$

Rewrite $- (x^{2} - 6 x - 20)$ as $- 1 (x^{2} - 6 x - 20)$ .

$\frac{- 1 (x^{2} - 6 x - 20)}{5 x (x + 4)} = 0$

Move the negative in front of the fraction.

$- \frac{x^{2} - 6 x - 20}{5 x (x + 4)} = 0$

Step 3

Set the numerator equal to zero.

$x^{2} - 6 x - 20 = 0$

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 $a = 1$ , $b = - 6$ , and $c = - 20$ into the quadratic formula and solve for $x$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (1 \cdot - 20)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot - 20}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 20$ .

Multiply $- 4$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot - 20}}{2 \cdot 1}$

Multiply $- 4$ by $- 20$ .

$x = \frac{6 \pm \sqrt{36 + 80}}{2 \cdot 1}$

Add $36$ and $80$ .

$x = \frac{6 \pm \sqrt{116}}{2 \cdot 1}$

Rewrite $116$ as $2^{2} \cdot 29$ .

Factor $4$ out of $116$ .

$x = \frac{6 \pm \sqrt{4 (29)}}{2 \cdot 1}$

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

$x = \frac{6 \pm \sqrt{2^{2} \cdot 29}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{6 \pm 2 \sqrt{29}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{6 \pm 2 \sqrt{29}}{2}$

Simplify $\frac{6 \pm 2 \sqrt{29}}{2}$ .

$x = 3 \pm \sqrt{29}$

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

Simplify the numerator.

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot - 20}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 20$ .

Multiply $- 4$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot - 20}}{2 \cdot 1}$

Multiply $- 4$ by $- 20$ .

$x = \frac{6 \pm \sqrt{36 + 80}}{2 \cdot 1}$

Add $36$ and $80$ .

$x = \frac{6 \pm \sqrt{116}}{2 \cdot 1}$

Rewrite $116$ as $2^{2} \cdot 29$ .

Factor $4$ out of $116$ .

$x = \frac{6 \pm \sqrt{4 (29)}}{2 \cdot 1}$

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

$x = \frac{6 \pm \sqrt{2^{2} \cdot 29}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{6 \pm 2 \sqrt{29}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{6 \pm 2 \sqrt{29}}{2}$

Simplify $\frac{6 \pm 2 \sqrt{29}}{2}$ .

$x = 3 \pm \sqrt{29}$

Change the $\pm$ to $+$ .

$x = 3 + \sqrt{29}$

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

Simplify the numerator.

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot - 20}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 20$ .

Multiply $- 4$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot - 20}}{2 \cdot 1}$

Multiply $- 4$ by $- 20$ .

$x = \frac{6 \pm \sqrt{36 + 80}}{2 \cdot 1}$

Add $36$ and $80$ .

$x = \frac{6 \pm \sqrt{116}}{2 \cdot 1}$

Rewrite $116$ as $2^{2} \cdot 29$ .

Factor $4$ out of $116$ .

$x = \frac{6 \pm \sqrt{4 (29)}}{2 \cdot 1}$

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

$x = \frac{6 \pm \sqrt{2^{2} \cdot 29}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{6 \pm 2 \sqrt{29}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{6 \pm 2 \sqrt{29}}{2}$

Simplify $\frac{6 \pm 2 \sqrt{29}}{2}$ .

$x = 3 \pm \sqrt{29}$

Change the $\pm$ to $-$ .

$x = 3 - \sqrt{29}$

The final answer is the combination of both solutions.

$x = 3 + \sqrt{29}, 3 - \sqrt{29}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = 3 + \sqrt{29}, 3 - \sqrt{29}$

Decimal Form:

$x = 8.38516480 \dots, - 2.38516480 \dots$

Solve by Factoring 1/x+1/(x+4)=1/5

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