Solve by Factoring (4x)/(x-2)+1/x=5

$\frac{4 x}{x - 2} + \frac{1}{x} = 5$

Step 1

Subtract $5$ from both sides of the equation.

$\frac{4 x}{x - 2} + \frac{1}{x} - 5 = 0$

Step 2

Simplify $\frac{4 x}{x - 2} + \frac{1}{x} - 5$ .

To write $\frac{4 x}{x - 2}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{4 x}{x - 2} \cdot \frac{x}{x} + \frac{1}{x} - 5 = 0$

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

$\frac{4 x}{x - 2} \cdot \frac{x}{x} + \frac{1}{x} \cdot \frac{x - 2}{x - 2} - 5 = 0$

Write each expression with a common denominator of $(x - 2) x$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{4 x}{x - 2}$ by $\frac{x}{x}$ .

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

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

$\frac{4 x \cdot x}{(x - 2) x} + \frac{x - 2}{x (x - 2)} - 5 = 0$

Reorder the factors of $(x - 2) x$ .

$\frac{4 x \cdot x}{x (x - 2)} + \frac{x - 2}{x (x - 2)} - 5 = 0$

Combine the numerators over the common denominator.

$\frac{4 x \cdot x + x - 2}{x (x - 2)} - 5 = 0$

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

Move $x$ .

$\frac{4 (x \cdot x) + x - 2}{x (x - 2)} - 5 = 0$

Multiply $x$ by $x$ .

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

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

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

Combine $- 5$ and $\frac{x (x - 2)}{x (x - 2)}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 2$ by $- 5$ .

$\frac{4 x^{2} + x - 2 - 5 x^{2} + 10 x}{x (x - 2)} = 0$

Subtract $5 x^{2}$ from $4 x^{2}$ .

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

Add $x$ and $10 x$ .

$\frac{- x^{2} + 11 x - 2}{x (x - 2)} = 0$

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

$\frac{- (x^{2}) + 11 x - 2}{x (x - 2)} = 0$

Factor $- 1$ out of $11 x$ .

$\frac{- (x^{2}) - (- 11 x) - 2}{x (x - 2)} = 0$

Factor $- 1$ out of $- (x^{2}) - (- 11 x)$ .

$\frac{- (x^{2} - 11 x) - 2}{x (x - 2)} = 0$

Rewrite $- 2$ as $- 1 (2)$ .

$\frac{- (x^{2} - 11 x) - 1 (2)}{x (x - 2)} = 0$

Factor $- 1$ out of $- (x^{2} - 11 x) - 1 (2)$ .

$\frac{- (x^{2} - 11 x + 2)}{x (x - 2)} = 0$

Rewrite $- (x^{2} - 11 x + 2)$ as $- 1 (x^{2} - 11 x + 2)$ .

$\frac{- 1 (x^{2} - 11 x + 2)}{x (x - 2)} = 0$

Move the negative in front of the fraction.

$- \frac{x^{2} - 11 x + 2}{x (x - 2)} = 0$

Step 3

Set the numerator equal to zero.

$x^{2} - 11 x + 2 = 0$

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 $a = 1$ , $b = - 11$ , and $c = 2$ into the quadratic formula and solve for $x$ .

$\frac{11 \pm \sqrt{{(- 11)}^{2} - 4 \cdot (1 \cdot 2)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $- 11$ to the power of $2$ .

$x = \frac{11 \pm \sqrt{121 - 4 \cdot 1 \cdot 2}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 2$ .

Multiply $- 4$ by $1$ .

$x = \frac{11 \pm \sqrt{121 - 4 \cdot 2}}{2 \cdot 1}$

Multiply $- 4$ by $2$ .

$x = \frac{11 \pm \sqrt{121 - 8}}{2 \cdot 1}$

Subtract $8$ from $121$ .

$x = \frac{11 \pm \sqrt{113}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{11 \pm \sqrt{113}}{2}$

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

Simplify the numerator.

Raise $- 11$ to the power of $2$ .

$x = \frac{11 \pm \sqrt{121 - 4 \cdot 1 \cdot 2}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 2$ .

Multiply $- 4$ by $1$ .

$x = \frac{11 \pm \sqrt{121 - 4 \cdot 2}}{2 \cdot 1}$

Multiply $- 4$ by $2$ .

$x = \frac{11 \pm \sqrt{121 - 8}}{2 \cdot 1}$

Subtract $8$ from $121$ .

$x = \frac{11 \pm \sqrt{113}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{11 \pm \sqrt{113}}{2}$

Change the $\pm$ to $+$ .

$x = \frac{11 + \sqrt{113}}{2}$

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

Simplify the numerator.

Raise $- 11$ to the power of $2$ .

$x = \frac{11 \pm \sqrt{121 - 4 \cdot 1 \cdot 2}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 2$ .

Multiply $- 4$ by $1$ .

$x = \frac{11 \pm \sqrt{121 - 4 \cdot 2}}{2 \cdot 1}$

Multiply $- 4$ by $2$ .

$x = \frac{11 \pm \sqrt{121 - 8}}{2 \cdot 1}$

Subtract $8$ from $121$ .

$x = \frac{11 \pm \sqrt{113}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{11 \pm \sqrt{113}}{2}$

Change the $\pm$ to $-$ .

$x = \frac{11 - \sqrt{113}}{2}$

The final answer is the combination of both solutions.

$x = \frac{11 + \sqrt{113}}{2}, \frac{11 - \sqrt{113}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{11 + \sqrt{113}}{2}, \frac{11 - \sqrt{113}}{2}$

Decimal Form:

$x = 10.81507290 \dots, 0.18492709 \dots$

Solve by Factoring (4x)/(x-2)+1/x=5

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