Trigonometry Examples

$9 x^{2} + 25 y^{2} = 225$

Step 1

Subtract $25 y^{2}$ from both sides of the equation.

$9 x^{2} = 225 - 25 y^{2}$

Step 2

Divide each term in $9 x^{2} = 225 - 25 y^{2}$ by $9$ and simplify.

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Step 2.1

Divide each term in $9 x^{2} = 225 - 25 y^{2}$ by $9$ .

$\frac{9 x^{2}}{9} = \frac{225}{9} + \frac{- 25 y^{2}}{9}$

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.

$\frac{9 x^{2}}{9} = \frac{225}{9} + \frac{- 25 y^{2}}{9}$

Step 2.2.1.2

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

$x^{2} = \frac{225}{9} + \frac{- 25 y^{2}}{9}$

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

Divide $225$ by $9$ .

$x^{2} = 25 + \frac{- 25 y^{2}}{9}$

Step 2.3.1.2

Move the negative in front of the fraction.

$x^{2} = 25 - \frac{25 y^{2}}{9}$

Step 3

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

$x = \pm \sqrt{25 - \frac{25 y^{2}}{9}}$

Step 4

Simplify $\pm \sqrt{25 - \frac{25 y^{2}}{9}}$ .

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Step 4.1

Write the expression using exponents.

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Step 4.1.1

Rewrite $25$ as $5^{2}$ .

$x = \pm \sqrt{5^{2} - \frac{25 y^{2}}{9}}$

Step 4.1.2

Rewrite $\frac{25 y^{2}}{9}$ as ${(\frac{5 y}{3})}^{2}$ .

$x = \pm \sqrt{5^{2} - {(\frac{5 y}{3})}^{2}}$

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 $a = 5$ and $b = \frac{5 y}{3}$ .

$x = \pm \sqrt{(5 + \frac{5 y}{3}) (5 - \frac{5 y}{3})}$

Step 4.3

To write $5$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = \pm \sqrt{(5 \cdot \frac{3}{3} + \frac{5 y}{3}) (5 - \frac{5 y}{3})}$

Step 4.4

Combine $5$ and $\frac{3}{3}$ .

$x = \pm \sqrt{(\frac{5 \cdot 3}{3} + \frac{5 y}{3}) (5 - \frac{5 y}{3})}$

Step 4.5

Combine the numerators over the common denominator.

$x = \pm \sqrt{\frac{5 \cdot 3 + 5 y}{3} (5 - \frac{5 y}{3})}$

Step 4.6

Factor $5$ out of $5 \cdot 3 + 5 y$ .

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Step 4.6.1

Factor $5$ out of $5 \cdot 3$ .

$x = \pm \sqrt{\frac{5 (3) + 5 y}{3} (5 - \frac{5 y}{3})}$

Step 4.6.2

Factor $5$ out of $5 y$ .

$x = \pm \sqrt{\frac{5 (3) + 5 (y)}{3} (5 - \frac{5 y}{3})}$

Step 4.6.3

Factor $5$ out of $5 (3) + 5 (y)$ .

$x = \pm \sqrt{\frac{5 (3 + y)}{3} (5 - \frac{5 y}{3})}$

Step 4.7

To write $5$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = \pm \sqrt{\frac{5 (3 + y)}{3} (5 \cdot \frac{3}{3} - \frac{5 y}{3})}$

Step 4.8

Combine $5$ and $\frac{3}{3}$ .

$x = \pm \sqrt{\frac{5 (3 + y)}{3} (\frac{5 \cdot 3}{3} - \frac{5 y}{3})}$

Step 4.9

Combine the numerators over the common denominator.

$x = \pm \sqrt{\frac{5 (3 + y)}{3} \cdot \frac{5 \cdot 3 - 5 y}{3}}$

Step 4.10

Factor $5$ out of $5 \cdot 3 - 5 y$ .

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Step 4.10.1

Factor $5$ out of $5 \cdot 3$ .

$x = \pm \sqrt{\frac{5 (3 + y)}{3} \cdot \frac{5 (3) - 5 y}{3}}$

Step 4.10.2

Factor $5$ out of $- 5 y$ .

$x = \pm \sqrt{\frac{5 (3 + y)}{3} \cdot \frac{5 (3) + 5 (- y)}{3}}$

Step 4.10.3

Factor $5$ out of $5 (3) + 5 (- y)$ .

$x = \pm \sqrt{\frac{5 (3 + y)}{3} \cdot \frac{5 (3 - y)}{3}}$

Step 4.11

Multiply $\frac{5 (3 + y)}{3}$ by $\frac{5 (3 - y)}{3}$ .

$x = \pm \sqrt{\frac{5 (3 + y) (5 (3 - y))}{3 \cdot 3}}$

Step 4.12

Multiply.

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Step 4.12.1

Multiply $5$ by $5$ .

$x = \pm \sqrt{\frac{25 (3 + y) (3 - y)}{3 \cdot 3}}$

Step 4.12.2

Multiply $3$ by $3$ .

$x = \pm \sqrt{\frac{25 (3 + y) (3 - y)}{9}}$

Step 4.13

Rewrite $\frac{25 (3 + y) (3 - y)}{9}$ as ${(\frac{5}{3})}^{2} ((3 + y) (3 - y))$ .

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Step 4.13.1

Factor the perfect power $5^{2}$ out of $25 (3 + y) (3 - y)$ .

$x = \pm \sqrt{\frac{5^{2} ((3 + y) (3 - y))}{9}}$

Step 4.13.2

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

$x = \pm \sqrt{\frac{5^{2} ((3 + y) (3 - y))}{3^{2} \cdot 1}}$

Step 4.13.3

Rearrange the fraction $\frac{5^{2} ((3 + y) (3 - y))}{3^{2} \cdot 1}$ .

$x = \pm \sqrt{{(\frac{5}{3})}^{2} ((3 + y) (3 - y))}$

Step 4.14

Pull terms out from under the radical.

$x = \pm \frac{5}{3} \sqrt{(3 + y) (3 - y)}$

Step 4.15

Combine $\frac{5}{3}$ and $\sqrt{(3 + y) (3 - y)}$ .

$x = \pm \frac{5 \sqrt{(3 + y) (3 - y)}}{3}$

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.

$x = \frac{5 \sqrt{(3 + y) (3 - y)}}{3}$

Step 5.2

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

$x = - \frac{5 \sqrt{(3 + y) (3 - y)}}{3}$

Step 5.3

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

$x = \frac{5 \sqrt{(3 + y) (3 - y)}}{3}$

$x = - \frac{5 \sqrt{(3 + y) (3 - y)}}{3}$

$x = \frac{5 \sqrt{(3 + y) (3 - y)}}{3}$

$x = - \frac{5 \sqrt{(3 + y) (3 - y)}}{3}$