Solve by Factoring 1/(x+6)+1/x=1/5

$\frac{1}{x + 6} + \frac{1}{x} = \frac{1}{5}$

Step 1

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

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

Step 2

Simplify $\frac{1}{x + 6} + \frac{1}{x} - \frac{1}{5}$ .

To write $\frac{1}{x + 6}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{1}{x + 6} \cdot \frac{x}{x} + \frac{1}{x} - \frac{1}{5} = 0$

To write $\frac{1}{x}$ as a fraction with a common denominator, multiply by $\frac{x + 6}{x + 6}$ .

$\frac{1}{x + 6} \cdot \frac{x}{x} + \frac{1}{x} \cdot \frac{x + 6}{x + 6} - \frac{1}{5} = 0$

Write each expression with a common denominator of $(x + 6) x$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{x + 6}$ by $\frac{x}{x}$ .

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

Multiply $\frac{1}{x}$ by $\frac{x + 6}{x + 6}$ .

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

Reorder the factors of $(x + 6) x$ .

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

Combine the numerators over the common denominator.

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

Add $x$ and $x$ .

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

Factor $2$ out of $2 x + 6$ .

Factor $2$ out of $2 x$ .

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

Factor $2$ out of $6$ .

$\frac{2 x + 2 \cdot 3}{x (x + 6)} - \frac{1}{5} = 0$

Factor $2$ out of $2 x + 2 \cdot 3$ .

$\frac{2 (x + 3)}{x (x + 6)} - \frac{1}{5} = 0$

To write $\frac{2 (x + 3)}{x (x + 6)}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{2 (x + 3)}{x (x + 6)} \cdot \frac{5}{5} - \frac{1}{5} = 0$

To write $- \frac{1}{5}$ as a fraction with a common denominator, multiply by $\frac{x (x + 6)}{x (x + 6)}$ .

$\frac{2 (x + 3)}{x (x + 6)} \cdot \frac{5}{5} - \frac{1}{5} \cdot \frac{x (x + 6)}{x (x + 6)} = 0$

Write each expression with a common denominator of $x (x + 6) \cdot 5$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{2 (x + 3)}{x (x + 6)}$ by $\frac{5}{5}$ .

$\frac{2 (x + 3) \cdot 5}{x (x + 6) \cdot 5} - \frac{1}{5} \cdot \frac{x (x + 6)}{x (x + 6)} = 0$

Multiply $\frac{1}{5}$ by $\frac{x (x + 6)}{x (x + 6)}$ .

$\frac{2 (x + 3) \cdot 5}{x (x + 6) \cdot 5} - \frac{x (x + 6)}{5 (x (x + 6))} = 0$

Reorder the factors of $x (x + 6) \cdot 5$ .

$\frac{2 (x + 3) \cdot 5}{5 x (x + 6)} - \frac{x (x + 6)}{5 (x (x + 6))} = 0$

Reorder the factors of $5 (x (x + 6))$ .

$\frac{2 (x + 3) \cdot 5}{5 x (x + 6)} - \frac{x (x + 6)}{5 x (x + 6)} = 0$

Combine the numerators over the common denominator.

$\frac{2 (x + 3) \cdot 5 - x (x + 6)}{5 x (x + 6)} = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{(2 x + 2 \cdot 3) \cdot 5 - x (x + 6)}{5 x (x + 6)} = 0$

Multiply $2$ by $3$ .

$\frac{(2 x + 6) \cdot 5 - x (x + 6)}{5 x (x + 6)} = 0$

Apply the distributive property.

$\frac{2 x \cdot 5 + 6 \cdot 5 - x (x + 6)}{5 x (x + 6)} = 0$

Multiply $5$ by $2$ .

$\frac{10 x + 6 \cdot 5 - x (x + 6)}{5 x (x + 6)} = 0$

Multiply $6$ by $5$ .

$\frac{10 x + 30 - x (x + 6)}{5 x (x + 6)} = 0$

Apply the distributive property.

$\frac{10 x + 30 - x \cdot x - x \cdot 6}{5 x (x + 6)} = 0$

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

Move $x$ .

$\frac{10 x + 30 - (x \cdot x) - x \cdot 6}{5 x (x + 6)} = 0$

Multiply $x$ by $x$ .

$\frac{10 x + 30 - x^{2} - x \cdot 6}{5 x (x + 6)} = 0$

Multiply $6$ by $- 1$ .

$\frac{10 x + 30 - x^{2} - 6 x}{5 x (x + 6)} = 0$

Subtract $6 x$ from $10 x$ .

$\frac{4 x + 30 - x^{2}}{5 x (x + 6)} = 0$

Reorder terms.

$\frac{- x^{2} + 4 x + 30}{5 x (x + 6)} = 0$

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

$\frac{- (x^{2}) + 4 x + 30}{5 x (x + 6)} = 0$

Factor $- 1$ out of $4 x$ .

$\frac{- (x^{2}) - (- 4 x) + 30}{5 x (x + 6)} = 0$

Factor $- 1$ out of $- (x^{2}) - (- 4 x)$ .

$\frac{- (x^{2} - 4 x) + 30}{5 x (x + 6)} = 0$

Rewrite $30$ as $- 1 (- 30)$ .

$\frac{- (x^{2} - 4 x) - 1 (- 30)}{5 x (x + 6)} = 0$

Factor $- 1$ out of $- (x^{2} - 4 x) - 1 (- 30)$ .

$\frac{- (x^{2} - 4 x - 30)}{5 x (x + 6)} = 0$

Rewrite $- (x^{2} - 4 x - 30)$ as $- 1 (x^{2} - 4 x - 30)$ .

$\frac{- 1 (x^{2} - 4 x - 30)}{5 x (x + 6)} = 0$

Move the negative in front of the fraction.

$- \frac{x^{2} - 4 x - 30}{5 x (x + 6)} = 0$

Step 3

Set the numerator equal to zero.

$x^{2} - 4 x - 30 = 0$

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 $a = 1$ , $b = - 4$ , and $c = - 30$ into the quadratic formula and solve for $x$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (1 \cdot - 30)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 30}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 30$ .

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot - 30}}{2 \cdot 1}$

Multiply $- 4$ by $- 30$ .

$x = \frac{4 \pm \sqrt{16 + 120}}{2 \cdot 1}$

Add $16$ and $120$ .

$x = \frac{4 \pm \sqrt{136}}{2 \cdot 1}$

Rewrite $136$ as $2^{2} \cdot 34$ .

Factor $4$ out of $136$ .

$x = \frac{4 \pm \sqrt{4 (34)}}{2 \cdot 1}$

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

$x = \frac{4 \pm \sqrt{2^{2} \cdot 34}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{4 \pm 2 \sqrt{34}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{4 \pm 2 \sqrt{34}}{2}$

Simplify $\frac{4 \pm 2 \sqrt{34}}{2}$ .

$x = 2 \pm \sqrt{34}$

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

Simplify the numerator.

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 30}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 30$ .

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot - 30}}{2 \cdot 1}$

Multiply $- 4$ by $- 30$ .

$x = \frac{4 \pm \sqrt{16 + 120}}{2 \cdot 1}$

Add $16$ and $120$ .

$x = \frac{4 \pm \sqrt{136}}{2 \cdot 1}$

Rewrite $136$ as $2^{2} \cdot 34$ .

Factor $4$ out of $136$ .

$x = \frac{4 \pm \sqrt{4 (34)}}{2 \cdot 1}$

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

$x = \frac{4 \pm \sqrt{2^{2} \cdot 34}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{4 \pm 2 \sqrt{34}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{4 \pm 2 \sqrt{34}}{2}$

Simplify $\frac{4 \pm 2 \sqrt{34}}{2}$ .

$x = 2 \pm \sqrt{34}$

Change the $\pm$ to $+$ .

$x = 2 + \sqrt{34}$

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

Simplify the numerator.

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 30}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 30$ .

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot - 30}}{2 \cdot 1}$

Multiply $- 4$ by $- 30$ .

$x = \frac{4 \pm \sqrt{16 + 120}}{2 \cdot 1}$

Add $16$ and $120$ .

$x = \frac{4 \pm \sqrt{136}}{2 \cdot 1}$

Rewrite $136$ as $2^{2} \cdot 34$ .

Factor $4$ out of $136$ .

$x = \frac{4 \pm \sqrt{4 (34)}}{2 \cdot 1}$

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

$x = \frac{4 \pm \sqrt{2^{2} \cdot 34}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{4 \pm 2 \sqrt{34}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{4 \pm 2 \sqrt{34}}{2}$

Simplify $\frac{4 \pm 2 \sqrt{34}}{2}$ .

$x = 2 \pm \sqrt{34}$

Change the $\pm$ to $-$ .

$x = 2 - \sqrt{34}$

The final answer is the combination of both solutions.

$x = 2 + \sqrt{34}, 2 - \sqrt{34}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = 2 + \sqrt{34}, 2 - \sqrt{34}$

Decimal Form:

$x = 7.83095189 \dots, - 3.83095189 \dots$

Solve by Factoring 1/(x+6)+1/x=1/5

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