Solve for x (2x+1)/x=1/(x-2)

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

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

$(2 x + 1) (x - 2) = x \cdot 1$

Solve the equation for $x$ .

Simplify $(2 x + 1) (x - 2)$ .

Start simplifying.

$0 + 0 + (2 x + 1) (x - 2) = x \cdot 1$

Simplify by adding zeros.

$(2 x + 1) (x - 2) = x \cdot 1$

Expand $(2 x + 1) (x - 2)$ using the FOIL Method.

Apply the distributive property.

$2 x (x - 2) + 1 (x - 2) = x \cdot 1$

Apply the distributive property.

$2 x \cdot x + 2 x \cdot - 2 + 1 (x - 2) = x \cdot 1$

Apply the distributive property.

$2 x \cdot x + 2 x \cdot - 2 + 1 x + 1 \cdot - 2 = x \cdot 1$

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

$2 (x \cdot x) + 2 x \cdot - 2 + 1 x + 1 \cdot - 2 = x \cdot 1$

Multiply $x$ by $x$ .

$2 x^{2} + 2 x \cdot - 2 + 1 x + 1 \cdot - 2 = x \cdot 1$

Multiply $- 2$ by $2$ .

$2 x^{2} - 4 x + 1 x + 1 \cdot - 2 = x \cdot 1$

Multiply $x$ by $1$ .

$2 x^{2} - 4 x + x + 1 \cdot - 2 = x \cdot 1$

Multiply $- 2$ by $1$ .

$2 x^{2} - 4 x + x - 2 = x \cdot 1$

Add $- 4 x$ and $x$ .

$2 x^{2} - 3 x - 2 = x \cdot 1$

Multiply $x$ by $1$ .

$2 x^{2} - 3 x - 2 = x$

Move all terms containing $x$ to the left side of the equation.

Subtract $x$ from both sides of the equation.

$2 x^{2} - 3 x - 2 - x = 0$

Subtract $x$ from $- 3 x$ .

$2 x^{2} - 4 x - 2 = 0$

Factor $2$ out of $2 x^{2} - 4 x - 2$ .

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

$2 (x^{2}) - 4 x - 2 = 0$

Factor $2$ out of $- 4 x$ .

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

Factor $2$ out of $- 2$ .

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

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

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

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

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

Divide each term in $2 (x^{2} - 2 x - 1) = 0$ by $2$ and simplify.

Divide each term in $2 (x^{2} - 2 x - 1) = 0$ by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x^{2} - 2 x - 1$ by $1$ .

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

Simplify the right side.

Divide $0$ by $2$ .

$x^{2} - 2 x - 1 = 0$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

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

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

Simplify.

Simplify the numerator.

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

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

Multiply $- 4$ by $1$ .

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

Multiply $- 4$ by $- 1$ .

$x = \frac{2 \pm \sqrt{4 + 4}}{2 \cdot 1}$

Add $4$ and $4$ .

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

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

$x = \frac{2 \pm \sqrt{4 (2)}}{2 \cdot 1}$

Rewrite $4$ as $2^{2}$ .

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

Pull terms out from under the radical.

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

Multiply $2$ by $1$ .

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

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

$x = 1 \pm \sqrt{2}$

The final answer is the combination of both solutions.

$x = 1 + \sqrt{2}, 1 - \sqrt{2}$

The result can be shown in multiple forms.

Exact Form:

$x = 1 + \sqrt{2}, 1 - \sqrt{2}$

Decimal Form:

$x = 2.41421356 \dots, - 0.41421356 \dots$

Solve for x (2x+1)/x=1/(x-2)

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