Calculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Dividing two negative values results in a positive value.

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

Step 2.2.2

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

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

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Divide $1$ by $- 1$ .

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

Step 2.3.1.2

Dividing two negative values results in a positive value.

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

Step 2.3.1.3

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

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

Step 3

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

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

Step 4

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

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Step 4.1

Simplify the expression.

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Step 4.1.1

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

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

Step 4.1.2

Reorder $- 1^{2}$ and $x^{2}$ .

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

Step 4.2

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 $b = 1$ .

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

Step 5

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

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Step 5.1

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

$y = \sqrt{(x + 1) (x - 1)}$

Step 5.2

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

$y = - \sqrt{(x + 1) (x - 1)}$

Step 5.3

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

$y = \sqrt{(x + 1) (x - 1)}$

$y = - \sqrt{(x + 1) (x - 1)}$

$y = \sqrt{(x + 1) (x - 1)}$

$y = - \sqrt{(x + 1) (x - 1)}$