Solve for y xy+y^2=1

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

Subtract $1$ from both sides of the equation.

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

Use the quadratic formula to find the solutions.

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

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

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

Simplify.

Simplify the numerator.

Multiply $- 4$ by $1$ .

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

Multiply $- 4$ by $- 1$ .

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

Multiply $2$ by $1$ .

$y = \frac{- x \pm \sqrt{x^{2} + 4}}{2}$

The final answer is the combination of both solutions.

$y = - \frac{x - \sqrt{x^{2} + 4}}{2}$

$y = - \frac{x + \sqrt{x^{2} + 4}}{2}$