Finite Math Examples

$9 x^{2} + 4 y^{2} - 36 = 0$

Step 1

Move all terms not containing $x$ to the right side of the equation.

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Step 1.1

Subtract $4 y^{2}$ from both sides of the equation.

$9 x^{2} - 36 = - 4 y^{2}$

Step 1.2

Add $36$ to both sides of the equation.

$9 x^{2} = - 4 y^{2} + 36$

Step 2

Divide each term in $9 x^{2} = - 4 y^{2} + 36$ by $9$ and simplify.

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Step 2.1

Divide each term in $9 x^{2} = - 4 y^{2} + 36$ by $9$ .

$\frac{9 x^{2}}{9} = \frac{- 4 y^{2}}{9} + \frac{36}{9}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $9$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{9 x^{2}}{9} = \frac{- 4 y^{2}}{9} + \frac{36}{9}$

Step 2.2.1.2

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

$x^{2} = \frac{- 4 y^{2}}{9} + \frac{36}{9}$

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

Move the negative in front of the fraction.

$x^{2} = - \frac{4 y^{2}}{9} + \frac{36}{9}$

Step 2.3.1.2

Divide $36$ by $9$ .

$x^{2} = - \frac{4 y^{2}}{9} + 4$

Step 3

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

$x = \pm \sqrt{- \frac{4 y^{2}}{9} + 4}$

Step 4

Simplify $\pm \sqrt{- \frac{4 y^{2}}{9} + 4}$ .

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Step 4.1

Factor $4$ out of $- \frac{4 y^{2}}{9} + 4$ .

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Step 4.1.1

Factor $4$ out of $- \frac{4 y^{2}}{9}$ .

$x = \pm \sqrt{4 (- \frac{y^{2}}{9}) + 4}$

Step 4.1.2

Factor $4$ out of $4$ .

$x = \pm \sqrt{4 (- \frac{y^{2}}{9}) + 4 (1)}$

Step 4.1.3

Factor $4$ out of $4 (- \frac{y^{2}}{9}) + 4 (1)$ .

$x = \pm \sqrt{4 (- \frac{y^{2}}{9} + 1)}$

Step 4.2

Simplify the expression.

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Step 4.2.1

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

$x = \pm \sqrt{4 (- \frac{y^{2}}{9} + 1^{2})}$

Step 4.2.2

Rewrite $\frac{y^{2}}{9}$ as ${(\frac{y}{3})}^{2}$ .

$x = \pm \sqrt{4 (- {(\frac{y}{3})}^{2} + 1^{2})}$

Step 4.2.3

Reorder $- {(\frac{y}{3})}^{2}$ and $1^{2}$ .

$x = \pm \sqrt{4 (1^{2} - {(\frac{y}{3})}^{2})}$

Step 4.3

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 $b = \frac{y}{3}$ .

$x = \pm \sqrt{4 (1 + \frac{y}{3}) (1 - \frac{y}{3})}$

Step 4.4

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

$x = \pm \sqrt{4 (\frac{3}{3} + \frac{y}{3}) (1 - \frac{y}{3})}$

Step 4.5

Combine the numerators over the common denominator.

$x = \pm \sqrt{4 \frac{3 + y}{3} (1 - \frac{y}{3})}$

Step 4.6

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

$x = \pm \sqrt{4 \frac{3 + y}{3} (\frac{3}{3} - \frac{y}{3})}$

Step 4.7

Combine the numerators over the common denominator.

$x = \pm \sqrt{4 \frac{3 + y}{3} \frac{3 - y}{3}}$

Step 4.8

Combine exponents.

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Step 4.8.1

Combine $4$ and $\frac{3 + y}{3}$ .

$x = \pm \sqrt{\frac{4 (3 + y)}{3} \cdot \frac{3 - y}{3}}$

Step 4.8.2

Multiply $\frac{4 (3 + y)}{3}$ by $\frac{3 - y}{3}$ .

$x = \pm \sqrt{\frac{4 (3 + y) (3 - y)}{3 \cdot 3}}$

Step 4.8.3

Multiply $3$ by $3$ .

$x = \pm \sqrt{\frac{4 (3 + y) (3 - y)}{9}}$

Step 4.9

Rewrite $\frac{4 (3 + y) (3 - y)}{9}$ as ${(\frac{2}{3})}^{2} ((3 + y) (3 - y))$ .

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Step 4.9.1

Factor the perfect power $2^{2}$ out of $4 (3 + y) (3 - y)$ .

$x = \pm \sqrt{\frac{2^{2} ((3 + y) (3 - y))}{9}}$

Step 4.9.2

Factor the perfect power $3^{2}$ out of $9$ .

$x = \pm \sqrt{\frac{2^{2} ((3 + y) (3 - y))}{3^{2} \cdot 1}}$

Step 4.9.3

Rearrange the fraction $\frac{2^{2} ((3 + y) (3 - y))}{3^{2} \cdot 1}$ .

$x = \pm \sqrt{{(\frac{2}{3})}^{2} ((3 + y) (3 - y))}$

Step 4.10

Pull terms out from under the radical.

$x = \pm \frac{2}{3} \sqrt{(3 + y) (3 - y)}$

Step 4.11

Combine $\frac{2}{3}$ and $\sqrt{(3 + y) (3 - y)}$ .

$x = \pm \frac{2 \sqrt{(3 + y) (3 - y)}}{3}$

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.

$x = \frac{2 \sqrt{(3 + y) (3 - y)}}{3}$

Step 5.2

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

$x = - \frac{2 \sqrt{(3 + y) (3 - y)}}{3}$

Step 5.3

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

$x = \frac{2 \sqrt{(3 + y) (3 - y)}}{3}$

$x = - \frac{2 \sqrt{(3 + y) (3 - y)}}{3}$

$x = \frac{2 \sqrt{(3 + y) (3 - y)}}{3}$

$x = - \frac{2 \sqrt{(3 + y) (3 - y)}}{3}$