Calculus Examples

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

Step 1

Solve the equation as $y$ in terms of $x$ .

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Dividing two negative values results in a positive value.

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

Step 1.2.2.2

Divide $\frac{y^{2}}{b^{2}}$ by $1$ .

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

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Divide $1$ by $- 1$ .

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

Step 1.2.3.1.2

Dividing two negative values results in a positive value.

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

Step 1.2.3.1.3

Divide $\frac{x^{2}}{a^{2}}$ by $1$ .

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

Step 1.3

Multiply both sides by $b^{2}$ .

$\frac{y^{2}}{b^{2}} b^{2} = (- 1 + \frac{x^{2}}{a^{2}}) b^{2}$

Step 1.4

Simplify.

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Step 1.4.1

Simplify the left side.

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Step 1.4.1.1

Cancel the common factor of $b^{2}$ .

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Step 1.4.1.1.1

Cancel the common factor.

$\frac{y^{2}}{b^{2}} b^{2} = (- 1 + \frac{x^{2}}{a^{2}}) b^{2}$

Step 1.4.1.1.2

Rewrite the expression.

$y^{2} = (- 1 + \frac{x^{2}}{a^{2}}) b^{2}$

Step 1.4.2

Simplify the right side.

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Step 1.4.2.1

Simplify $(- 1 + \frac{x^{2}}{a^{2}}) b^{2}$ .

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Step 1.4.2.1.1

Apply the distributive property.

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

Step 1.4.2.1.2

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

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

Step 1.4.2.1.3

Combine $\frac{x^{2}}{a^{2}}$ and $b^{2}$ .

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

Step 1.4.2.1.4

Simplify with commuting.

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Step 1.4.2.1.4.1

Reorder $x^{2}$ and $b^{2}$ .

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

Step 1.4.2.1.4.2

Reorder $- b^{2}$ and $\frac{b^{2} x^{2}}{a^{2}}$ .

$y^{2} = \frac{b^{2} x^{2}}{a^{2}} - b^{2}$

Step 1.5

Solve for $y$ .

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Step 1.5.1

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

$y = \pm \sqrt{\frac{b^{2} x^{2}}{a^{2}} - b^{2}}$

Step 1.5.2

Simplify $\pm \sqrt{\frac{b^{2} x^{2}}{a^{2}} - b^{2}}$ .

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Step 1.5.2.1

Factor $b^{2}$ out of $\frac{b^{2} x^{2}}{a^{2}} - b^{2}$ .

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Step 1.5.2.1.1

Factor $b^{2}$ out of $\frac{b^{2} x^{2}}{a^{2}}$ .

$y = \pm \sqrt{b^{2} \frac{x^{2}}{a^{2}} - b^{2}}$

Step 1.5.2.1.2

Factor $b^{2}$ out of $- b^{2}$ .

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

Step 1.5.2.1.3

Factor $b^{2}$ out of $b^{2} \frac{x^{2}}{a^{2}} + b^{2} \cdot - 1$ .

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

Step 1.5.2.2

Rewrite $\frac{x^{2}}{a^{2}}$ as ${(\frac{x}{a})}^{2}$ .

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

Step 1.5.2.3

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

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

Step 1.5.2.4

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

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

Step 1.5.2.5

Write $1$ as a fraction with a common denominator.

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

Step 1.5.2.6

Combine the numerators over the common denominator.

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

Step 1.5.2.7

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{a}{a}$ .

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

Step 1.5.2.8

Combine $- 1$ and $\frac{a}{a}$ .

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

Step 1.5.2.9

Combine the numerators over the common denominator.

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

Step 1.5.2.10

Combine exponents.

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Step 1.5.2.10.1

Combine $b^{2}$ and $\frac{x + a}{a}$ .

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

Step 1.5.2.10.2

Multiply $\frac{b^{2} (x + a)}{a}$ by $\frac{x - a}{a}$ .

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

Step 1.5.2.10.3

Raise $a$ to the power of $1$ .

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

Step 1.5.2.10.4

Raise $a$ to the power of $1$ .

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

Step 1.5.2.10.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.5.2.10.6

Add $1$ and $1$ .

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

Step 1.5.2.11

Rewrite $\frac{b^{2} (x + a) (x - a)}{a^{2}}$ as ${(\frac{b}{a})}^{2} ((x + a) (x - a))$ .

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Step 1.5.2.11.1

Factor the perfect power $b^{2}$ out of $b^{2} (x + a) (x - a)$ .

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

Step 1.5.2.11.2

Factor the perfect power $a^{2}$ out of $a^{2}$ .

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

Step 1.5.2.11.3

Rearrange the fraction $\frac{b^{2} ((x + a) (x - a))}{a^{2} \cdot 1}$ .

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

Step 1.5.2.12

Pull terms out from under the radical.

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

Step 1.5.2.13

Combine $\frac{b}{a}$ and $\sqrt{(x + a) (x - a)}$ .

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

Step 1.5.3

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

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Step 1.5.3.1

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