Solve for f e^2+f^2=1

$e^{2} + f^{2} = 1$

Step 1

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

$f^{2} = 1 - e^{2}$

Step 2

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

$f = \pm \sqrt{1 - e^{2}}$

Step 3

Simplify $\pm \sqrt{1 - e^{2}}$ .

Rewrite $1$ as $1^{2}$ .

$f = \pm \sqrt{1^{2} - e^{2}}$

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

$f = \pm \sqrt{(1 + e) (1 - e)}$

Step 4

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.

$f = \sqrt{(1 + e) (1 - e)}$

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

$f = - \sqrt{(1 + e) (1 - e)}$

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

$f = \sqrt{(1 + e) (1 - e)}, - \sqrt{(1 + e) (1 - e)}$