Calculus Examples

$16 x^{2} + 25 y^{2} = 400$

Step 1

Subtract $16 x^{2}$ from both sides of the equation.

$25 y^{2} = 400 - 16 x^{2}$

Step 2

Divide each term in $25 y^{2} = 400 - 16 x^{2}$ by $25$ and simplify.

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Step 2.1

Divide each term in $25 y^{2} = 400 - 16 x^{2}$ by $25$ .

$\frac{25 y^{2}}{25} = \frac{400}{25} + \frac{- 16 x^{2}}{25}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $25$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{25 y^{2}}{25} = \frac{400}{25} + \frac{- 16 x^{2}}{25}$

Step 2.2.1.2

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

$y^{2} = \frac{400}{25} + \frac{- 16 x^{2}}{25}$

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 $400$ by $25$ .

$y^{2} = 16 + \frac{- 16 x^{2}}{25}$

Step 2.3.1.2

Move the negative in front of the fraction.

$y^{2} = 16 - \frac{16 x^{2}}{25}$

Step 3

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

$y = \pm \sqrt{16 - \frac{16 x^{2}}{25}}$

Step 4

Simplify $\pm \sqrt{16 - \frac{16 x^{2}}{25}}$ .

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Step 4.1

Write the expression using exponents.

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Step 4.1.1

Rewrite $16$ as $4^{2}$ .

$y = \pm \sqrt{4^{2} - \frac{16 x^{2}}{25}}$

Step 4.1.2

Rewrite $\frac{16 x^{2}}{25}$ as ${(\frac{4 x}{5})}^{2}$ .

$y = \pm \sqrt{4^{2} - {(\frac{4 x}{5})}^{2}}$

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 $a = 4$ and $b = \frac{4 x}{5}$ .

$y = \pm \sqrt{(4 + \frac{4 x}{5}) (4 - \frac{4 x}{5})}$

Step 4.3

To write $4$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y = \pm \sqrt{(4 \cdot \frac{5}{5} + \frac{4 x}{5}) (4 - \frac{4 x}{5})}$

Step 4.4

Combine $4$ and $\frac{5}{5}$ .

$y = \pm \sqrt{(\frac{4 \cdot 5}{5} + \frac{4 x}{5}) (4 - \frac{4 x}{5})}$

Step 4.5

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{4 \cdot 5 + 4 x}{5} (4 - \frac{4 x}{5})}$

Step 4.6

Factor $4$ out of $4 \cdot 5 + 4 x$ .

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Step 4.6.1

Factor $4$ out of $4 \cdot 5$ .

$y = \pm \sqrt{\frac{4 (5) + 4 x}{5} (4 - \frac{4 x}{5})}$

Step 4.6.2

Factor $4$ out of $4 x$ .

$y = \pm \sqrt{\frac{4 (5) + 4 (x)}{5} (4 - \frac{4 x}{5})}$

Step 4.6.3

Factor $4$ out of $4 (5) + 4 (x)$ .

$y = \pm \sqrt{\frac{4 (5 + x)}{5} (4 - \frac{4 x}{5})}$

Step 4.7

To write $4$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y = \pm \sqrt{\frac{4 (5 + x)}{5} (4 \cdot \frac{5}{5} - \frac{4 x}{5})}$

Step 4.8

Combine $4$ and $\frac{5}{5}$ .

$y = \pm \sqrt{\frac{4 (5 + x)}{5} (\frac{4 \cdot 5}{5} - \frac{4 x}{5})}$

Step 4.9

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{4 (5 + x)}{5} \cdot \frac{4 \cdot 5 - 4 x}{5}}$

Step 4.10

Factor $4$ out of $4 \cdot 5 - 4 x$ .

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Step 4.10.1

Factor $4$ out of $4 \cdot 5$ .

$y = \pm \sqrt{\frac{4 (5 + x)}{5} \cdot \frac{4 (5) - 4 x}{5}}$

Step 4.10.2

Factor $4$ out of $- 4 x$ .

$y = \pm \sqrt{\frac{4 (5 + x)}{5} \cdot \frac{4 (5) + 4 (- x)}{5}}$

Step 4.10.3

Factor $4$ out of $4 (5) + 4 (- x)$ .

$y = \pm \sqrt{\frac{4 (5 + x)}{5} \cdot \frac{4 (5 - x)}{5}}$

Step 4.11

Multiply $\frac{4 (5 + x)}{5}$ by $\frac{4 (5 - x)}{5}$ .

$y = \pm \sqrt{\frac{4 (5 + x) (4 (5 - x))}{5 \cdot 5}}$

Step 4.12

Multiply.

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Step 4.12.1

Multiply $4$ by $4$ .

$y = \pm \sqrt{\frac{16 (5 + x) (5 - x)}{5 \cdot 5}}$

Step 4.12.2

Multiply $5$ by $5$ .

$y = \pm \sqrt{\frac{16 (5 + x) (5 - x)}{25}}$

Step 4.13

Rewrite $\frac{16 (5 + x) (5 - x)}{25}$ as ${(\frac{2^{2}}{5})}^{2} ((5 + x) (5 - x))$ .

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Step 4.13.1

Factor the perfect power ${(2^{2})}^{2}$ out of $16 (5 + x) (5 - x)$ .

$y = \pm \sqrt{\frac{{(2^{2})}^{2} ((5 + x) (5 - x))}{25}}$

Step 4.13.2

Factor the perfect power $5^{2}$ out of $25$ .

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

Step 4.13.3

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

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

Step 4.14

Pull terms out from under the radical.

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

Step 4.15

Raise $2$ to the power of $2$ .

$y = \pm \frac{4}{5} \sqrt{(5 + x) (5 - x)}$

Step 4.16

Combine $\frac{4}{5}$ and $\sqrt{(5 + x) (5 - x)}$ .

$y = \pm \frac{4 \sqrt{(5 + x) (5 - x)}}{5}$

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.

$y = \frac{4 \sqrt{(5 + x) (5 - x)}}{5}$

Step 5.2

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

$y = - \frac{4 \sqrt{(5 + x) (5 - x)}}{5}$

Step 5.3

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

$y = \frac{4 \sqrt{(5 + x) (5 - x)}}{5}$

$y = - \frac{4 \sqrt{(5 + x) (5 - x)}}{5}$

$y = \frac{4 \sqrt{(5 + x) (5 - x)}}{5}$

$y = - \frac{4 \sqrt{(5 + x) (5 - x)}}{5}$