Finite Math Examples

$4 y^{2} + x^{2} = 144$

Step 1

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

$4 y^{2} = 144 - x^{2}$

Step 2

Divide each term in $4 y^{2} = 144 - x^{2}$ by $4$ and simplify.

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Step 2.1

Divide each term in $4 y^{2} = 144 - x^{2}$ by $4$ .

$\frac{4 y^{2}}{4} = \frac{144}{4} + \frac{- x^{2}}{4}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $4$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{4 y^{2}}{4} = \frac{144}{4} + \frac{- x^{2}}{4}$

Step 2.2.1.2

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

$y^{2} = \frac{144}{4} + \frac{- x^{2}}{4}$

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 $144$ by $4$ .

$y^{2} = 36 + \frac{- x^{2}}{4}$

Step 2.3.1.2

Move the negative in front of the fraction.

$y^{2} = 36 - \frac{x^{2}}{4}$

Step 3

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

$y = \pm \sqrt{36 - \frac{x^{2}}{4}}$

Step 4

Simplify $\pm \sqrt{36 - \frac{x^{2}}{4}}$ .

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Step 4.1

Write the expression using exponents.

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Step 4.1.1

Rewrite $36$ as $6^{2}$ .

$y = \pm \sqrt{6^{2} - \frac{x^{2}}{4}}$

Step 4.1.2

Rewrite $\frac{x^{2}}{4}$ as ${(\frac{x}{2})}^{2}$ .

$y = \pm \sqrt{6^{2} - {(\frac{x}{2})}^{2}}$

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 $a = 6$ and $b = \frac{x}{2}$ .

$y = \pm \sqrt{(6 + \frac{x}{2}) (6 - \frac{x}{2})}$

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.

$y = \sqrt{(6 + \frac{x}{2}) (6 - \frac{x}{2})}$

Step 5.2

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

$y = - \sqrt{(6 + \frac{x}{2}) (6 - \frac{x}{2})}$

Step 5.3

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

$y = \sqrt{(6 + \frac{x}{2}) (6 - \frac{x}{2})}$

$y = - \sqrt{(6 + \frac{x}{2}) (6 - \frac{x}{2})}$

$y = \sqrt{(6 + \frac{x}{2}) (6 - \frac{x}{2})}$

$y = - \sqrt{(6 + \frac{x}{2}) (6 - \frac{x}{2})}$

Step 6

To write $6$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \sqrt{(6 \cdot \frac{2}{2} + \frac{x}{2}) (6 - \frac{x}{2})}$

$y = - \sqrt{(6 + \frac{x}{2}) (6 - \frac{x}{2})}$

Step 7

Combine $6$ and $\frac{2}{2}$ .

$y = \sqrt{(\frac{6 \cdot 2}{2} + \frac{x}{2}) (6 - \frac{x}{2})}$

$y = - \sqrt{(6 + \frac{x}{2}) (6 - \frac{x}{2})}$

Step 8

Combine the numerators over the common denominator.

$y = \sqrt{\frac{6 \cdot 2 + x}{2} \cdot (6 - \frac{x}{2})}$

$y = - \sqrt{(6 + \frac{x}{2}) (6 - \frac{x}{2})}$

Step 9

Multiply $6$ by $2$ .

$y = \sqrt{\frac{12 + x}{2} \cdot (6 - \frac{x}{2})}$

$y = - \sqrt{(6 + \frac{x}{2}) (6 - \frac{x}{2})}$

Step 10

To write $6$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \sqrt{\frac{12 + x}{2} \cdot (6 \cdot \frac{2}{2} - \frac{x}{2})}$

$y = - \sqrt{(6 + \frac{x}{2}) (6 - \frac{x}{2})}$

Step 11

Combine $6$ and $\frac{2}{2}$ .

$y = \sqrt{\frac{12 + x}{2} \cdot (\frac{6 \cdot 2}{2} - \frac{x}{2})}$

$y = - \sqrt{(6 + \frac{x}{2}) (6 - \frac{x}{2})}$

Step 12

Combine the numerators over the common denominator.

$y = \sqrt{\frac{12 + x}{2} \cdot \frac{6 \cdot 2 - x}{2}}$

$y = - \sqrt{(6 + \frac{x}{2}) (6 - \frac{x}{2})}$

Step 13

Multiply $6$ by $2$ .

$y = \sqrt{\frac{12 + x}{2} \cdot \frac{12 - x}{2}}$

$y = - \sqrt{(6 + \frac{x}{2}) (6 - \frac{x}{2})}$

Step 14

Multiply $\frac{12 + x}{2}$ by $\frac{12 - x}{2}$ .

$y = \sqrt{\frac{(12 + x) (12 - x)}{2 \cdot 2}}$

$y = - \sqrt{(6 + \frac{x}{2}) (6 - \frac{x}{2})}$

Step 15

Multiply $2$ by $2$ .

$y = \sqrt{\frac{(12 + x) (12 - x)}{4}}$

$y = - \sqrt{(6 + \frac{x}{2}) (6 - \frac{x}{2})}$

Step 16

Rewrite $\frac{(12 + x) (12 - x)}{4}$ as ${(\frac{1}{2})}^{2} ((12 + x) (12 - x))$ .

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Step 16.1

Factor the perfect power $1^{2}$ out of $(12 + x) (12 - x)$ .

$y = \sqrt{\frac{1^{2} ((12 + x) (12 - x))}{4}}$

$y = - \sqrt{(6 + \frac{x}{2}) (6 - \frac{x}{2})}$

Step 16.2

Factor the perfect power $2^{2}$ out of $4$ .

$y = \sqrt{\frac{1^{2} ((12 + x) (12 - x))}{2^{2} \cdot 1}}$

$y = - \sqrt{(6 + \frac{x}{2}) (6 - \frac{x}{2})}$

Step 16.3

Rearrange the fraction $\frac{1^{2} ((12 + x) (12 - x))}{2^{2} \cdot 1}$ .

$y = \sqrt{{(\frac{1}{2})}^{2} ((12 + x) (12 - x))}$

$y = - \sqrt{(6 + \frac{x}{2}) (6 - \frac{x}{2})}$

$y = \sqrt{{(\frac{1}{2})}^{2} ((12 + x) (12 - x))}$

$y = - \sqrt{(6 + \frac{x}{2}) (6 - \frac{x}{2})}$

Step 17

Pull terms out from under the radical.

$y = \frac{1}{2} \cdot \sqrt{(12 + x) (12 - x)}$

$y = - \sqrt{(6 + \frac{x}{2}) (6 - \frac{x}{2})}$

Step 18

Combine $\frac{1}{2}$ and $\sqrt{(12 + x) (12 - x)}$ .

$y = \frac{\sqrt{(12 + x) (12 - x)}}{2}$

$y = - \sqrt{(6 + \frac{x}{2}) (6 - \frac{x}{2})}$

Step 19

To write $6$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{\sqrt{(12 + x) (12 - x)}}{2}$

$y = - \sqrt{(6 \cdot \frac{2}{2} + \frac{x}{2}) (6 - \frac{x}{2})}$

Step 20

Combine $6$ and $\frac{2}{2}$ .

$y = \frac{\sqrt{(12 + x) (12 - x)}}{2}$

$y = - \sqrt{(\frac{6 \cdot 2}{2} + \frac{x}{2}) (6 - \frac{x}{2})}$

Step 21

Combine the numerators over the common denominator.

$y = \frac{\sqrt{(12 + x) (12 - x)}}{2}$

$y = - \sqrt{\frac{6 \cdot 2 + x}{2} \cdot (6 - \frac{x}{2})}$

Step 22

Multiply $6$ by $2$ .

$y = \frac{\sqrt{(12 + x) (12 - x)}}{2}$

$y = - \sqrt{\frac{12 + x}{2} \cdot (6 - \frac{x}{2})}$

Step 23

To write $6$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{\sqrt{(12 + x) (12 - x)}}{2}$

$y = - \sqrt{\frac{12 + x}{2} \cdot (6 \cdot \frac{2}{2} - \frac{x}{2})}$

Step 24

Combine $6$ and $\frac{2}{2}$ .

$y = \frac{\sqrt{(12 + x) (12 - x)}}{2}$

$y = - \sqrt{\frac{12 + x}{2} \cdot (\frac{6 \cdot 2}{2} - \frac{x}{2})}$

Step 25

Combine the numerators over the common denominator.

$y = \frac{\sqrt{(12 + x) (12 - x)}}{2}$

$y = - \sqrt{\frac{12 + x}{2} \cdot \frac{6 \cdot 2 - x}{2}}$

Step 26

Multiply $6$ by $2$ .

$y = \frac{\sqrt{(12 + x) (12 - x)}}{2}$

$y = - \sqrt{\frac{12 + x}{2} \cdot \frac{12 - x}{2}}$

Step 27

Multiply $\frac{12 + x}{2}$ by $\frac{12 - x}{2}$ .

$y = \frac{\sqrt{(12 + x) (12 - x)}}{2}$

$y = - \sqrt{\frac{(12 + x) (12 - x)}{2 \cdot 2}}$

Step 28

Multiply $2$ by $2$ .

$y = \frac{\sqrt{(12 + x) (12 - x)}}{2}$

$y = - \sqrt{\frac{(12 + x) (12 - x)}{4}}$

Step 29

Rewrite $\frac{(12 + x) (12 - x)}{4}$ as ${(\frac{1}{2})}^{2} ((12 + x) (12 - x))$ .

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Step 29.1

Factor the perfect power $1^{2}$ out of $(12 + x) (12 - x)$ .

$y = \frac{\sqrt{(12 + x) (12 - x)}}{2}$

$y = - \sqrt{\frac{1^{2} ((12 + x) (12 - x))}{4}}$

Step 29.2

Factor the perfect power $2^{2}$ out of $4$ .

$y = \frac{\sqrt{(12 + x) (12 - x)}}{2}$

$y = - \sqrt{\frac{1^{2} ((12 + x) (12 - x))}{2^{2} \cdot 1}}$

Step 29.3

Rearrange the fraction $\frac{1^{2} ((12 + x) (12 - x))}{2^{2} \cdot 1}$ .

$y = \frac{\sqrt{(12 + x) (12 - x)}}{2}$

$y = - \sqrt{{(\frac{1}{2})}^{2} ((12 + x) (12 - x))}$

$y = \frac{\sqrt{(12 + x) (12 - x)}}{2}$

$y = - \sqrt{{(\frac{1}{2})}^{2} ((12 + x) (12 - x))}$

Step 30

Pull terms out from under the radical.

$y = \frac{\sqrt{(12 + x) (12 - x)}}{2}$

$y = - (\frac{1}{2} \cdot \sqrt{(12 + x) (12 - x)})$

Step 31

Combine $\frac{1}{2}$ and $\sqrt{(12 + x) (12 - x)}$ .

$y = \frac{\sqrt{(12 + x) (12 - x)}}{2}$

$y = - \frac{\sqrt{(12 + x) (12 - x)}}{2}$

Step 32

The given equation $y = \frac{\sqrt{(12 + x) (12 - x)}}{2}, - \frac{\sqrt{(12 + x) (12 - x)}}{2}$ can not be written as $y = k x$ , so $y$ doesn't vary directly with $x$ .

$y$ doesn't vary directly with $x$