Solve by Factoring 7/x+1/(x-5)=1

$\frac{7}{x} + \frac{1}{x - 5} = 1$

Step 1

Subtract $1$ from both sides of the equation.

$\frac{7}{x} + \frac{1}{x - 5} - 1 = 0$

Step 2

Simplify $\frac{7}{x} + \frac{1}{x - 5} - 1$ .

To write $\frac{7}{x}$ as a fraction with a common denominator, multiply by $\frac{x - 5}{x - 5}$ .

$\frac{7}{x} \cdot \frac{x - 5}{x - 5} + \frac{1}{x - 5} - 1 = 0$

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

$\frac{7}{x} \cdot \frac{x - 5}{x - 5} + \frac{1}{x - 5} \cdot \frac{x}{x} - 1 = 0$

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

Multiply $\frac{7}{x}$ by $\frac{x - 5}{x - 5}$ .

$\frac{7 (x - 5)}{x (x - 5)} + \frac{1}{x - 5} \cdot \frac{x}{x} - 1 = 0$

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

$\frac{7 (x - 5)}{x (x - 5)} + \frac{x}{(x - 5) x} - 1 = 0$

Reorder the factors of $(x - 5) x$ .

$\frac{7 (x - 5)}{x (x - 5)} + \frac{x}{x (x - 5)} - 1 = 0$

Combine the numerators over the common denominator.

$\frac{7 (x - 5) + x}{x (x - 5)} - 1 = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{7 x + 7 \cdot - 5 + x}{x (x - 5)} - 1 = 0$

Multiply $7$ by $- 5$ .

$\frac{7 x - 35 + x}{x (x - 5)} - 1 = 0$

Add $7 x$ and $x$ .

$\frac{8 x - 35}{x (x - 5)} - 1 = 0$

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

$\frac{8 x - 35}{x (x - 5)} - 1 \cdot \frac{x (x - 5)}{x (x - 5)} = 0$

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

$\frac{8 x - 35}{x (x - 5)} + \frac{- (x (x - 5))}{x (x - 5)} = 0$

Combine the numerators over the common denominator.

$\frac{8 x - 35 - (x (x - 5))}{x (x - 5)} = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{8 x - 35 - x \cdot x - x \cdot - 5}{x (x - 5)} = 0$

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

Move $x$ .

$\frac{8 x - 35 - (x \cdot x) - x \cdot - 5}{x (x - 5)} = 0$

Multiply $x$ by $x$ .

$\frac{8 x - 35 - x^{2} - x \cdot - 5}{x (x - 5)} = 0$

Multiply $- 5$ by $- 1$ .

$\frac{8 x - 35 - x^{2} + 5 x}{x (x - 5)} = 0$

Add $8 x$ and $5 x$ .

$\frac{13 x - 35 - x^{2}}{x (x - 5)} = 0$

Reorder terms.

$\frac{- x^{2} + 13 x - 35}{x (x - 5)} = 0$

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

$\frac{- (x^{2}) + 13 x - 35}{x (x - 5)} = 0$

Factor $- 1$ out of $13 x$ .

$\frac{- (x^{2}) - (- 13 x) - 35}{x (x - 5)} = 0$

Factor $- 1$ out of $- (x^{2}) - (- 13 x)$ .

$\frac{- (x^{2} - 13 x) - 35}{x (x - 5)} = 0$

Rewrite $- 35$ as $- 1 (35)$ .

$\frac{- (x^{2} - 13 x) - 1 (35)}{x (x - 5)} = 0$

Factor $- 1$ out of $- (x^{2} - 13 x) - 1 (35)$ .

$\frac{- (x^{2} - 13 x + 35)}{x (x - 5)} = 0$

Rewrite $- (x^{2} - 13 x + 35)$ as $- 1 (x^{2} - 13 x + 35)$ .

$\frac{- 1 (x^{2} - 13 x + 35)}{x (x - 5)} = 0$

Move the negative in front of the fraction.

$- \frac{x^{2} - 13 x + 35}{x (x - 5)} = 0$

Step 3

Set the numerator equal to zero.

$x^{2} - 13 x + 35 = 0$

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 $a = 1$ , $b = - 13$ , and $c = 35$ into the quadratic formula and solve for $x$ .

$\frac{13 \pm \sqrt{{(- 13)}^{2} - 4 \cdot (1 \cdot 35)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $- 13$ to the power of $2$ .

$x = \frac{13 \pm \sqrt{169 - 4 \cdot 1 \cdot 35}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 35$ .

Multiply $- 4$ by $1$ .

$x = \frac{13 \pm \sqrt{169 - 4 \cdot 35}}{2 \cdot 1}$

Multiply $- 4$ by $35$ .

$x = \frac{13 \pm \sqrt{169 - 140}}{2 \cdot 1}$

Subtract $140$ from $169$ .

$x = \frac{13 \pm \sqrt{29}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{13 \pm \sqrt{29}}{2}$

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

Simplify the numerator.

Raise $- 13$ to the power of $2$ .

$x = \frac{13 \pm \sqrt{169 - 4 \cdot 1 \cdot 35}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 35$ .

Multiply $- 4$ by $1$ .

$x = \frac{13 \pm \sqrt{169 - 4 \cdot 35}}{2 \cdot 1}$

Multiply $- 4$ by $35$ .

$x = \frac{13 \pm \sqrt{169 - 140}}{2 \cdot 1}$

Subtract $140$ from $169$ .

$x = \frac{13 \pm \sqrt{29}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{13 \pm \sqrt{29}}{2}$

Change the $\pm$ to $+$ .

$x = \frac{13 + \sqrt{29}}{2}$

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

Simplify the numerator.

Raise $- 13$ to the power of $2$ .

$x = \frac{13 \pm \sqrt{169 - 4 \cdot 1 \cdot 35}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 35$ .

Multiply $- 4$ by $1$ .

$x = \frac{13 \pm \sqrt{169 - 4 \cdot 35}}{2 \cdot 1}$

Multiply $- 4$ by $35$ .

$x = \frac{13 \pm \sqrt{169 - 140}}{2 \cdot 1}$

Subtract $140$ from $169$ .

$x = \frac{13 \pm \sqrt{29}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{13 \pm \sqrt{29}}{2}$

Change the $\pm$ to $-$ .

$x = \frac{13 - \sqrt{29}}{2}$

The final answer is the combination of both solutions.

$x = \frac{13 + \sqrt{29}}{2}, \frac{13 - \sqrt{29}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{13 + \sqrt{29}}{2}, \frac{13 - \sqrt{29}}{2}$

Decimal Form:

$x = 9.19258240 \dots, 3.80741759 \dots$

Solve by Factoring 7/x+1/(x-5)=1

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