Solve by Factoring 9/x+1/(x-1)=9

$\frac{9}{x} + \frac{1}{x - 1} = 9$

Step 1

Subtract $9$ from both sides of the equation.

$\frac{9}{x} + \frac{1}{x - 1} - 9 = 0$

Step 2

Simplify $\frac{9}{x} + \frac{1}{x - 1} - 9$ .

To write $\frac{9}{x}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$\frac{9}{x} \cdot \frac{x - 1}{x - 1} + \frac{1}{x - 1} - 9 = 0$

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

$\frac{9}{x} \cdot \frac{x - 1}{x - 1} + \frac{1}{x - 1} \cdot \frac{x}{x} - 9 = 0$

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

Multiply $\frac{9}{x}$ by $\frac{x - 1}{x - 1}$ .

$\frac{9 (x - 1)}{x (x - 1)} + \frac{1}{x - 1} \cdot \frac{x}{x} - 9 = 0$

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

$\frac{9 (x - 1)}{x (x - 1)} + \frac{x}{(x - 1) x} - 9 = 0$

Reorder the factors of $(x - 1) x$ .

$\frac{9 (x - 1)}{x (x - 1)} + \frac{x}{x (x - 1)} - 9 = 0$

Combine the numerators over the common denominator.

$\frac{9 (x - 1) + x}{x (x - 1)} - 9 = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{9 x + 9 \cdot - 1 + x}{x (x - 1)} - 9 = 0$

Multiply $9$ by $- 1$ .

$\frac{9 x - 9 + x}{x (x - 1)} - 9 = 0$

Add $9 x$ and $x$ .

$\frac{10 x - 9}{x (x - 1)} - 9 = 0$

To write $- 9$ as a fraction with a common denominator, multiply by $\frac{x (x - 1)}{x (x - 1)}$ .

$\frac{10 x - 9}{x (x - 1)} - 9 \cdot \frac{x (x - 1)}{x (x - 1)} = 0$

Combine $- 9$ and $\frac{x (x - 1)}{x (x - 1)}$ .

$\frac{10 x - 9}{x (x - 1)} + \frac{- 9 (x (x - 1))}{x (x - 1)} = 0$

Combine the numerators over the common denominator.

$\frac{10 x - 9 - 9 (x (x - 1))}{x (x - 1)} = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{10 x - 9 - 9 x \cdot x - 9 x \cdot - 1}{x (x - 1)} = 0$

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

Move $x$ .

$\frac{10 x - 9 - 9 (x \cdot x) - 9 x \cdot - 1}{x (x - 1)} = 0$

Multiply $x$ by $x$ .

$\frac{10 x - 9 - 9 x^{2} - 9 x \cdot - 1}{x (x - 1)} = 0$

Multiply $- 1$ by $- 9$ .

$\frac{10 x - 9 - 9 x^{2} + 9 x}{x (x - 1)} = 0$

Add $10 x$ and $9 x$ .

$\frac{19 x - 9 - 9 x^{2}}{x (x - 1)} = 0$

Reorder terms.

$\frac{- 9 x^{2} + 19 x - 9}{x (x - 1)} = 0$

Factor $- 1$ out of $- 9 x^{2}$ .

$\frac{- (9 x^{2}) + 19 x - 9}{x (x - 1)} = 0$

Factor $- 1$ out of $19 x$ .

$\frac{- (9 x^{2}) - (- 19 x) - 9}{x (x - 1)} = 0$

Factor $- 1$ out of $- (9 x^{2}) - (- 19 x)$ .

$\frac{- (9 x^{2} - 19 x) - 9}{x (x - 1)} = 0$

Rewrite $- 9$ as $- 1 (9)$ .

$\frac{- (9 x^{2} - 19 x) - 1 (9)}{x (x - 1)} = 0$

Factor $- 1$ out of $- (9 x^{2} - 19 x) - 1 (9)$ .

$\frac{- (9 x^{2} - 19 x + 9)}{x (x - 1)} = 0$

Rewrite $- (9 x^{2} - 19 x + 9)$ as $- 1 (9 x^{2} - 19 x + 9)$ .

$\frac{- 1 (9 x^{2} - 19 x + 9)}{x (x - 1)} = 0$

Move the negative in front of the fraction.

$- \frac{9 x^{2} - 19 x + 9}{x (x - 1)} = 0$

Step 3

Set the numerator equal to zero.

$9 x^{2} - 19 x + 9 = 0$

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 $a = 9$ , $b = - 19$ , and $c = 9$ into the quadratic formula and solve for $x$ .

$\frac{19 \pm \sqrt{{(- 19)}^{2} - 4 \cdot (9 \cdot 9)}}{2 \cdot 9}$

Simplify.

Simplify the numerator.

Raise $- 19$ to the power of $2$ .

$x = \frac{19 \pm \sqrt{361 - 4 \cdot 9 \cdot 9}}{2 \cdot 9}$

Multiply $- 4 \cdot 9 \cdot 9$ .

Multiply $- 4$ by $9$ .

$x = \frac{19 \pm \sqrt{361 - 36 \cdot 9}}{2 \cdot 9}$

Multiply $- 36$ by $9$ .

$x = \frac{19 \pm \sqrt{361 - 324}}{2 \cdot 9}$

Subtract $324$ from $361$ .

$x = \frac{19 \pm \sqrt{37}}{2 \cdot 9}$

Multiply $2$ by $9$ .

$x = \frac{19 \pm \sqrt{37}}{18}$

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

Simplify the numerator.

Raise $- 19$ to the power of $2$ .

$x = \frac{19 \pm \sqrt{361 - 4 \cdot 9 \cdot 9}}{2 \cdot 9}$

Multiply $- 4 \cdot 9 \cdot 9$ .

Multiply $- 4$ by $9$ .

$x = \frac{19 \pm \sqrt{361 - 36 \cdot 9}}{2 \cdot 9}$

Multiply $- 36$ by $9$ .

$x = \frac{19 \pm \sqrt{361 - 324}}{2 \cdot 9}$

Subtract $324$ from $361$ .

$x = \frac{19 \pm \sqrt{37}}{2 \cdot 9}$

Multiply $2$ by $9$ .

$x = \frac{19 \pm \sqrt{37}}{18}$

Change the $\pm$ to $+$ .

$x = \frac{19 + \sqrt{37}}{18}$

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

Simplify the numerator.

Raise $- 19$ to the power of $2$ .

$x = \frac{19 \pm \sqrt{361 - 4 \cdot 9 \cdot 9}}{2 \cdot 9}$

Multiply $- 4 \cdot 9 \cdot 9$ .

Multiply $- 4$ by $9$ .

$x = \frac{19 \pm \sqrt{361 - 36 \cdot 9}}{2 \cdot 9}$

Multiply $- 36$ by $9$ .

$x = \frac{19 \pm \sqrt{361 - 324}}{2 \cdot 9}$

Subtract $324$ from $361$ .

$x = \frac{19 \pm \sqrt{37}}{2 \cdot 9}$

Multiply $2$ by $9$ .

$x = \frac{19 \pm \sqrt{37}}{18}$

Change the $\pm$ to $-$ .

$x = \frac{19 - \sqrt{37}}{18}$

The final answer is the combination of both solutions.

$x = \frac{19 + \sqrt{37}}{18}, \frac{19 - \sqrt{37}}{18}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{19 + \sqrt{37}}{18}, \frac{19 - \sqrt{37}}{18}$

Decimal Form:

$x = 1.39348680 \dots, 0.71762430 \dots$

Solve by Factoring 9/x+1/(x-1)=9

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