Algebra Examples

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

Step 1

Add $\frac{y^{2}}{b^{2}}$ to both sides of the equation.

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

Step 2

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

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

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

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

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Step 3.1.1.1

Cancel the common factor.

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

Step 3.1.1.2

Rewrite the expression.

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

Step 3.2

Simplify the right side.

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Step 3.2.1

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

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Step 3.2.1.1

Apply the distributive property.

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

Step 3.2.1.2

Multiply $a^{2}$ by $1$ .

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

Step 3.2.1.3

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

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

Step 3.2.1.4

Simplify with commuting.

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Step 3.2.1.4.1

Reorder $y^{2}$ and $a^{2}$ .

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

Step 3.2.1.4.2

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

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

Step 4

Solve for $x$ .

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Step 4.1

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

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

Step 4.2

Simplify $\pm \sqrt{\frac{a^{2} y^{2}}{b^{2}} + a^{2}}$ .

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Step 4.2.1

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

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Step 4.2.1.1

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

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

Step 4.2.1.2

Multiply by $1$ .

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

Step 4.2.1.3

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

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

Step 4.2.2

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

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

Step 4.2.3

Combine the numerators over the common denominator.

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

Step 4.2.4

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

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

Step 4.2.5

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

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Step 4.2.5.1

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

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

Step 4.2.5.2

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

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

Step 4.2.6

Pull terms out from under the radical.

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

Step 4.2.7

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

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

Step 4.3

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

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Step 4.3.1

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

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

Step 4.3.2

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

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

Step 4.3.3

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

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

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

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

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

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

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