Solve for x y=x+1/x

$y = x + \frac{1}{x}$

Rewrite the equation as $x + \frac{1}{x} = y$ .

$x + \frac{1}{x} = y$

Find the LCD of the terms in the equation.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, x, 1$

The LCM of one and any expression is the expression.

$x$

Multiply each term in $x + \frac{1}{x} = y$ by $x$ to eliminate the fractions.

Multiply each term in $x + \frac{1}{x} = y$ by $x$ .

$x \cdot x + \frac{1}{x} x = y x$

Simplify the left side.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + \frac{1}{x} x = y x$

Cancel the common factor of $x$ .

Cancel the common factor.

$x^{2} + \frac{1}{x} x = y x$

Rewrite the expression.

$x^{2} + 1 = y x$

Solve the equation.

Subtract $y x$ from both sides of the equation.

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

Use the quadratic formula to find the solutions.

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

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

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

Simplify.

Simplify the numerator.

Rewrite $4 \cdot 1 \cdot 1$ as ${(2 \cdot 1 \cdot 1)}^{2}$ .

$x = \frac{y \pm \sqrt{{(- y)}^{2} - {(2 \cdot 1 \cdot 1)}^{2}}}{2 \cdot 1}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = - y$ and $b = 2 \cdot 1 \cdot 1$ .

$x = \frac{y \pm \sqrt{(- y + 2 \cdot 1 \cdot 1) (- y - (2 \cdot 1 \cdot 1))}}{2 \cdot 1}$

Simplify.

Multiply $2$ by $1$ .

$x = \frac{y \pm \sqrt{(- y + 2 \cdot 1) (- y - (2 \cdot 1 \cdot 1))}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{y \pm \sqrt{(- y + 2) (- y - (2 \cdot 1 \cdot 1))}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{y \pm \sqrt{(- y + 2) (- y - (2 \cdot 1))}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{y \pm \sqrt{(- y + 2) (- y - 1 \cdot 2)}}{2 \cdot 1}$

Multiply $- 1$ by $2$ .

$x = \frac{y \pm \sqrt{(- y + 2) (- y - 2)}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{y \pm \sqrt{(- y + 2) (- y - 2)}}{2}$

The final answer is the combination of both solutions.

$x = \frac{y + \sqrt{(- y + 2) (- y - 2)}}{2}$

$x = \frac{y - \sqrt{(- y + 2) (- y - 2)}}{2}$

$x = \frac{y + \sqrt{(- y + 2) (- y - 2)}}{2}$

$x = \frac{y - \sqrt{(- y + 2) (- y - 2)}}{2}$

Solve for x y=x+1/x

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