Solve for y x^2-y^2=z^2

$x^{2} - y^{2} = z^{2}$

Subtract $x^{2}$ from both sides of the equation.

$- y^{2} = z^{2} - x^{2}$

Divide each term in $- y^{2} = z^{2} - x^{2}$ by $- 1$ and simplify.

Divide each term in $- y^{2} = z^{2} - x^{2}$ by $- 1$ .

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

Divide $y^{2}$ by $1$ .

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

Simplify the right side.

Simplify each term.

Move the negative one from the denominator of $\frac{z^{2}}{- 1}$ .

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

Rewrite $- 1 \cdot z^{2}$ as $- z^{2}$ .

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

Dividing two negative values results in a positive value.

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

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

$y^{2} = - z^{2} + x^{2}$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

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

Simplify $\pm \sqrt{- z^{2} + x^{2}}$ .

Reorder $- z^{2}$ and $x^{2}$ .

$y = \pm \sqrt{x^{2} - z^{2}}$

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

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

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

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

Next, use the negative value of the $\pm$ to find the second solution.

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

The complete solution is the result of both the positive and negative portions of the solution.

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

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

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

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