Solve by Factoring 6/x+2/(x-1)=9

$\frac{6}{x} + \frac{2}{x - 1} = 9$

Step 1

Subtract $9$ from both sides of the equation.

$\frac{6}{x} + \frac{2}{x - 1} - 9 = 0$

Step 2

Simplify $\frac{6}{x} + \frac{2}{x - 1} - 9$ .

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

$\frac{6}{x} \cdot \frac{x - 1}{x - 1} + \frac{2}{x - 1} - 9 = 0$

To write $\frac{2}{x - 1}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{6}{x} \cdot \frac{x - 1}{x - 1} + \frac{2}{x - 1} \cdot \frac{x}{x} - 9 = 0$

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

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

$\frac{6 (x - 1)}{x (x - 1)} + \frac{2}{x - 1} \cdot \frac{x}{x} - 9 = 0$

Multiply $\frac{2}{x - 1}$ by $\frac{x}{x}$ .

$\frac{6 (x - 1)}{x (x - 1)} + \frac{2 x}{(x - 1) x} - 9 = 0$

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

$\frac{6 (x - 1)}{x (x - 1)} + \frac{2 x}{x (x - 1)} - 9 = 0$

Combine the numerators over the common denominator.

$\frac{6 (x - 1) + 2 x}{x (x - 1)} - 9 = 0$

Simplify the numerator.

Factor $2$ out of $6 (x - 1) + 2 x$ .

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

$\frac{2 (3 (x - 1)) + 2 x}{x (x - 1)} - 9 = 0$

Factor $2$ out of $2 x$ .

$\frac{2 (3 (x - 1)) + 2 (x)}{x (x - 1)} - 9 = 0$

Factor $2$ out of $2 (3 (x - 1)) + 2 (x)$ .

$\frac{2 (3 (x - 1) + x)}{x (x - 1)} - 9 = 0$

Apply the distributive property.

$\frac{2 (3 x + 3 \cdot - 1 + x)}{x (x - 1)} - 9 = 0$

Multiply $3$ by $- 1$ .

$\frac{2 (3 x - 3 + x)}{x (x - 1)} - 9 = 0$

Add $3 x$ and $x$ .

$\frac{2 (4 x - 3)}{x (x - 1)} - 9 = 0$

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

$\frac{2 (4 x - 3)}{x (x - 1)} - 9 \cdot \frac{x (x - 1)}{x (x - 1)} = 0$

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

$\frac{2 (4 x - 3)}{x (x - 1)} + \frac{- 9 (x (x - 1))}{x (x - 1)} = 0$

Combine the numerators over the common denominator.

$\frac{2 (4 x - 3) - 9 (x (x - 1))}{x (x - 1)} = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{2 (4 x) + 2 \cdot - 3 - 9 x (x - 1)}{x (x - 1)} = 0$

Multiply $4$ by $2$ .

$\frac{8 x + 2 \cdot - 3 - 9 x (x - 1)}{x (x - 1)} = 0$

Multiply $2$ by $- 3$ .

$\frac{8 x - 6 - 9 x (x - 1)}{x (x - 1)} = 0$

Apply the distributive property.

$\frac{8 x - 6 - 9 x \cdot x - 9 x \cdot - 1}{x (x - 1)} = 0$

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

Move $x$ .

$\frac{8 x - 6 - 9 (x \cdot x) - 9 x \cdot - 1}{x (x - 1)} = 0$

Multiply $x$ by $x$ .

$\frac{8 x - 6 - 9 x^{2} - 9 x \cdot - 1}{x (x - 1)} = 0$

Multiply $- 1$ by $- 9$ .

$\frac{8 x - 6 - 9 x^{2} + 9 x}{x (x - 1)} = 0$

Add $8 x$ and $9 x$ .

$\frac{17 x - 6 - 9 x^{2}}{x (x - 1)} = 0$

Reorder terms.

$\frac{- 9 x^{2} + 17 x - 6}{x (x - 1)} = 0$

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

$\frac{- (9 x^{2}) + 17 x - 6}{x (x - 1)} = 0$

Factor $- 1$ out of $17 x$ .

$\frac{- (9 x^{2}) - (- 17 x) - 6}{x (x - 1)} = 0$

Factor $- 1$ out of $- (9 x^{2}) - (- 17 x)$ .

$\frac{- (9 x^{2} - 17 x) - 6}{x (x - 1)} = 0$

Rewrite $- 6$ as $- 1 (6)$ .

$\frac{- (9 x^{2} - 17 x) - 1 (6)}{x (x - 1)} = 0$

Factor $- 1$ out of $- (9 x^{2} - 17 x) - 1 (6)$ .

$\frac{- (9 x^{2} - 17 x + 6)}{x (x - 1)} = 0$

Rewrite $- (9 x^{2} - 17 x + 6)$ as $- 1 (9 x^{2} - 17 x + 6)$ .

$\frac{- 1 (9 x^{2} - 17 x + 6)}{x (x - 1)} = 0$

Move the negative in front of the fraction.

$- \frac{9 x^{2} - 17 x + 6}{x (x - 1)} = 0$

Step 3

Set the numerator equal to zero.

$9 x^{2} - 17 x + 6 = 0$

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 $a = 9$ , $b = - 17$ , and $c = 6$ into the quadratic formula and solve for $x$ .

$\frac{17 \pm \sqrt{{(- 17)}^{2} - 4 \cdot (9 \cdot 6)}}{2 \cdot 9}$

Simplify.

Simplify the numerator.

Raise $- 17$ to the power of $2$ .

$x = \frac{17 \pm \sqrt{289 - 4 \cdot 9 \cdot 6}}{2 \cdot 9}$

Multiply $- 4 \cdot 9 \cdot 6$ .

Multiply $- 4$ by $9$ .

$x = \frac{17 \pm \sqrt{289 - 36 \cdot 6}}{2 \cdot 9}$

Multiply $- 36$ by $6$ .

$x = \frac{17 \pm \sqrt{289 - 216}}{2 \cdot 9}$

Subtract $216$ from $289$ .

$x = \frac{17 \pm \sqrt{73}}{2 \cdot 9}$

Multiply $2$ by $9$ .

$x = \frac{17 \pm \sqrt{73}}{18}$

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

Simplify the numerator.

Raise $- 17$ to the power of $2$ .

$x = \frac{17 \pm \sqrt{289 - 4 \cdot 9 \cdot 6}}{2 \cdot 9}$

Multiply $- 4 \cdot 9 \cdot 6$ .

Multiply $- 4$ by $9$ .

$x = \frac{17 \pm \sqrt{289 - 36 \cdot 6}}{2 \cdot 9}$

Multiply $- 36$ by $6$ .

$x = \frac{17 \pm \sqrt{289 - 216}}{2 \cdot 9}$

Subtract $216$ from $289$ .

$x = \frac{17 \pm \sqrt{73}}{2 \cdot 9}$

Multiply $2$ by $9$ .

$x = \frac{17 \pm \sqrt{73}}{18}$

Change the $\pm$ to $+$ .

$x = \frac{17 + \sqrt{73}}{18}$

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

Simplify the numerator.

Raise $- 17$ to the power of $2$ .

$x = \frac{17 \pm \sqrt{289 - 4 \cdot 9 \cdot 6}}{2 \cdot 9}$

Multiply $- 4 \cdot 9 \cdot 6$ .

Multiply $- 4$ by $9$ .

$x = \frac{17 \pm \sqrt{289 - 36 \cdot 6}}{2 \cdot 9}$

Multiply $- 36$ by $6$ .

$x = \frac{17 \pm \sqrt{289 - 216}}{2 \cdot 9}$

Subtract $216$ from $289$ .

$x = \frac{17 \pm \sqrt{73}}{2 \cdot 9}$

Multiply $2$ by $9$ .

$x = \frac{17 \pm \sqrt{73}}{18}$

Change the $\pm$ to $-$ .

$x = \frac{17 - \sqrt{73}}{18}$

The final answer is the combination of both solutions.

$x = \frac{17 + \sqrt{73}}{18}, \frac{17 - \sqrt{73}}{18}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{17 + \sqrt{73}}{18}, \frac{17 - \sqrt{73}}{18}$

Decimal Form:

$x = 1.41911131 \dots, 0.46977756 \dots$

Solve by Factoring 6/x+2/(x-1)=9

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