Algebra Examples

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

Step 1

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

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

Step 3.2.1.4

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

Simplify the expression.

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Step 4.2.2.1

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

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

Step 4.2.2.2

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

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

Step 4.2.2.3

Reorder $- {(\frac{y}{b})}^{2}$ and $1^{2}$ .

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

Step 4.2.3

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 $b = \frac{y}{b}$ .

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

Step 4.2.4

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

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

Step 4.2.5

Combine the numerators over the common denominator.

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

Step 4.2.6

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

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

Step 4.2.7

Combine the numerators over the common denominator.

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

Step 4.2.8

Combine exponents.

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Step 4.2.8.1

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

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

Step 4.2.8.2

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

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

Step 4.2.8.3

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

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

Step 4.2.8.4

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

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

Step 4.2.8.5

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

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

Step 4.2.8.6

Add $1$ and $1$ .

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

Step 4.2.9

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

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Step 4.2.9.1

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

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

Step 4.2.9.2

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

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

Step 4.2.9.3

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

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

Step 4.2.10

Pull terms out from under the radical.

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

Step 4.2.11

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

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

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.

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

Step 4.3.2

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

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

Step 4.3.3

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

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

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

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

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

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

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