Solve by Factoring 1/x+1/(x+2)=1/7

$\frac{1}{x} + \frac{1}{x + 2} = \frac{1}{7}$

Step 1

Subtract $\frac{1}{7}$ from both sides of the equation.

$\frac{1}{x} + \frac{1}{x + 2} - \frac{1}{7} = 0$

Step 2

Simplify $\frac{1}{x} + \frac{1}{x + 2} - \frac{1}{7}$ .

Find the common denominator.

Multiply $\frac{1}{x}$ by $\frac{(x + 2) \cdot 7}{(x + 2) \cdot 7}$ .

$\frac{1}{x} \cdot \frac{(x + 2) \cdot 7}{(x + 2) \cdot 7} + \frac{1}{x + 2} - \frac{1}{7} = 0$

Multiply $\frac{1}{x}$ by $\frac{(x + 2) \cdot 7}{(x + 2) \cdot 7}$ .

$\frac{(x + 2) \cdot 7}{x ((x + 2) \cdot 7)} + \frac{1}{x + 2} - \frac{1}{7} = 0$

Multiply $\frac{1}{x + 2}$ by $\frac{x \cdot 7}{x \cdot 7}$ .

$\frac{(x + 2) \cdot 7}{x ((x + 2) \cdot 7)} + \frac{1}{x + 2} \cdot \frac{x \cdot 7}{x \cdot 7} - \frac{1}{7} = 0$

Multiply $\frac{1}{x + 2}$ by $\frac{x \cdot 7}{x \cdot 7}$ .

$\frac{(x + 2) \cdot 7}{x ((x + 2) \cdot 7)} + \frac{x \cdot 7}{(x + 2) (x \cdot 7)} - \frac{1}{7} = 0$

Multiply $\frac{1}{7}$ by $\frac{x (x + 2)}{x (x + 2)}$ .

$\frac{(x + 2) \cdot 7}{x ((x + 2) \cdot 7)} + \frac{x \cdot 7}{(x + 2) (x \cdot 7)} - (\frac{1}{7} \cdot \frac{x (x + 2)}{x (x + 2)}) = 0$

Multiply $\frac{1}{7}$ by $\frac{x (x + 2)}{x (x + 2)}$ .

$\frac{(x + 2) \cdot 7}{x ((x + 2) \cdot 7)} + \frac{x \cdot 7}{(x + 2) (x \cdot 7)} - \frac{x (x + 2)}{7 (x (x + 2))} = 0$

Reorder the factors of $x ((x + 2) \cdot 7)$ .

$\frac{(x + 2) \cdot 7}{7 x (x + 2)} + \frac{x \cdot 7}{(x + 2) (x \cdot 7)} - \frac{x (x + 2)}{7 (x (x + 2))} = 0$

Reorder the factors of $(x + 2) (x \cdot 7)$ .

$\frac{(x + 2) \cdot 7}{7 x (x + 2)} + \frac{x \cdot 7}{7 x (x + 2)} - \frac{x (x + 2)}{7 (x (x + 2))} = 0$

Reorder the factors of $7 (x (x + 2))$ .

$\frac{(x + 2) \cdot 7}{7 x (x + 2)} + \frac{x \cdot 7}{7 x (x + 2)} - \frac{x (x + 2)}{7 x (x + 2)} = 0$

Combine the numerators over the common denominator.

$\frac{(x + 2) \cdot 7 + x \cdot 7 - x (x + 2)}{7 x (x + 2)} = 0$

Simplify each term.

Apply the distributive property.

$\frac{x \cdot 7 + 2 \cdot 7 + x \cdot 7 - x (x + 2)}{7 x (x + 2)} = 0$

Move $7$ to the left of $x$ .

$\frac{7 \cdot x + 2 \cdot 7 + x \cdot 7 - x (x + 2)}{7 x (x + 2)} = 0$

Multiply $2$ by $7$ .

$\frac{7 \cdot x + 14 + x \cdot 7 - x (x + 2)}{7 x (x + 2)} = 0$

Move $7$ to the left of $x$ .

$\frac{7 x + 14 + 7 \cdot x - x (x + 2)}{7 x (x + 2)} = 0$

Apply the distributive property.

$\frac{7 x + 14 + 7 x - x \cdot x - x \cdot 2}{7 x (x + 2)} = 0$

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

Move $x$ .

$\frac{7 x + 14 + 7 x - (x \cdot x) - x \cdot 2}{7 x (x + 2)} = 0$

Multiply $x$ by $x$ .

$\frac{7 x + 14 + 7 x - x^{2} - x \cdot 2}{7 x (x + 2)} = 0$

Multiply $2$ by $- 1$ .

$\frac{7 x + 14 + 7 x - x^{2} - 2 x}{7 x (x + 2)} = 0$

Add $7 x$ and $7 x$ .

$\frac{14 x + 14 - x^{2} - 2 x}{7 x (x + 2)} = 0$

Subtract $2 x$ from $14 x$ .

$\frac{12 x + 14 - x^{2}}{7 x (x + 2)} = 0$

Reorder terms.

$\frac{- x^{2} + 12 x + 14}{7 x (x + 2)} = 0$

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

$\frac{- (x^{2}) + 12 x + 14}{7 x (x + 2)} = 0$

Factor $- 1$ out of $12 x$ .

$\frac{- (x^{2}) - (- 12 x) + 14}{7 x (x + 2)} = 0$

Factor $- 1$ out of $- (x^{2}) - (- 12 x)$ .

$\frac{- (x^{2} - 12 x) + 14}{7 x (x + 2)} = 0$

Rewrite $14$ as $- 1 (- 14)$ .

$\frac{- (x^{2} - 12 x) - 1 (- 14)}{7 x (x + 2)} = 0$

Factor $- 1$ out of $- (x^{2} - 12 x) - 1 (- 14)$ .

$\frac{- (x^{2} - 12 x - 14)}{7 x (x + 2)} = 0$

Rewrite $- (x^{2} - 12 x - 14)$ as $- 1 (x^{2} - 12 x - 14)$ .

$\frac{- 1 (x^{2} - 12 x - 14)}{7 x (x + 2)} = 0$

Move the negative in front of the fraction.

$- \frac{x^{2} - 12 x - 14}{7 x (x + 2)} = 0$

Step 3

Set the numerator equal to zero.

$x^{2} - 12 x - 14 = 0$

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 $a = 1$ , $b = - 12$ , and $c = - 14$ into the quadratic formula and solve for $x$ .

$\frac{12 \pm \sqrt{{(- 12)}^{2} - 4 \cdot (1 \cdot - 14)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $- 12$ to the power of $2$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 1 \cdot - 14}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 14$ .

Multiply $- 4$ by $1$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot - 14}}{2 \cdot 1}$

Multiply $- 4$ by $- 14$ .

$x = \frac{12 \pm \sqrt{144 + 56}}{2 \cdot 1}$

Add $144$ and $56$ .

$x = \frac{12 \pm \sqrt{200}}{2 \cdot 1}$

Rewrite $200$ as $10^{2} \cdot 2$ .

Factor $100$ out of $200$ .

$x = \frac{12 \pm \sqrt{100 (2)}}{2 \cdot 1}$

Rewrite $100$ as $10^{2}$ .

$x = \frac{12 \pm \sqrt{10^{2} \cdot 2}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{12 \pm 10 \sqrt{2}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{12 \pm 10 \sqrt{2}}{2}$

Simplify $\frac{12 \pm 10 \sqrt{2}}{2}$ .

$x = 6 \pm 5 \sqrt{2}$

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

Simplify the numerator.

Raise $- 12$ to the power of $2$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 1 \cdot - 14}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 14$ .

Multiply $- 4$ by $1$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot - 14}}{2 \cdot 1}$

Multiply $- 4$ by $- 14$ .

$x = \frac{12 \pm \sqrt{144 + 56}}{2 \cdot 1}$

Add $144$ and $56$ .

$x = \frac{12 \pm \sqrt{200}}{2 \cdot 1}$

Rewrite $200$ as $10^{2} \cdot 2$ .

Factor $100$ out of $200$ .

$x = \frac{12 \pm \sqrt{100 (2)}}{2 \cdot 1}$

Rewrite $100$ as $10^{2}$ .

$x = \frac{12 \pm \sqrt{10^{2} \cdot 2}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{12 \pm 10 \sqrt{2}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{12 \pm 10 \sqrt{2}}{2}$

Simplify $\frac{12 \pm 10 \sqrt{2}}{2}$ .

$x = 6 \pm 5 \sqrt{2}$

Change the $\pm$ to $+$ .

$x = 6 + 5 \sqrt{2}$

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

Simplify the numerator.

Raise $- 12$ to the power of $2$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 1 \cdot - 14}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 14$ .

Multiply $- 4$ by $1$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot - 14}}{2 \cdot 1}$

Multiply $- 4$ by $- 14$ .

$x = \frac{12 \pm \sqrt{144 + 56}}{2 \cdot 1}$

Add $144$ and $56$ .

$x = \frac{12 \pm \sqrt{200}}{2 \cdot 1}$

Rewrite $200$ as $10^{2} \cdot 2$ .

Factor $100$ out of $200$ .

$x = \frac{12 \pm \sqrt{100 (2)}}{2 \cdot 1}$

Rewrite $100$ as $10^{2}$ .

$x = \frac{12 \pm \sqrt{10^{2} \cdot 2}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{12 \pm 10 \sqrt{2}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{12 \pm 10 \sqrt{2}}{2}$

Simplify $\frac{12 \pm 10 \sqrt{2}}{2}$ .

$x = 6 \pm 5 \sqrt{2}$

Change the $\pm$ to $-$ .

$x = 6 - 5 \sqrt{2}$

The final answer is the combination of both solutions.

$x = 6 + 5 \sqrt{2}, 6 - 5 \sqrt{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = 6 + 5 \sqrt{2}, 6 - 5 \sqrt{2}$

Decimal Form:

$x = 13.07106781 \dots, - 1.07106781 \dots$

Solve by Factoring 1/x+1/(x+2)=1/7

Related Questions

Support LCMGCF.com by disabling your adblocker.