Precalculus Examples

$\frac{x^{2}}{4} + \frac{y^{2}}{25} = 1$

Step 1

Subtract $\frac{x^{2}}{4}$ from both sides of the equation.

$\frac{y^{2}}{25} = 1 - \frac{x^{2}}{4}$

Step 2

Multiply both sides of the equation by $25$ .

$25 \frac{y^{2}}{25} = 25 (1 - \frac{x^{2}}{4})$

Step 3

Simplify both sides of the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Cancel the common factor of $25$ .

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Step 3.1.1.1

Cancel the common factor.

$25 \frac{y^{2}}{25} = 25 (1 - \frac{x^{2}}{4})$

Step 3.1.1.2

Rewrite the expression.

$y^{2} = 25 (1 - \frac{x^{2}}{4})$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $25 (1 - \frac{x^{2}}{4})$ .

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Step 3.2.1.1

Apply the distributive property.

$y^{2} = 25 \cdot 1 + 25 (- \frac{x^{2}}{4})$

Step 3.2.1.2

Multiply $25$ by $1$ .

$y^{2} = 25 + 25 (- \frac{x^{2}}{4})$

Step 3.2.1.3

Multiply $25 (- \frac{x^{2}}{4})$ .

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Step 3.2.1.3.1

Multiply $- 1$ by $25$ .

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

Step 3.2.1.3.2

Combine $- 25$ and $\frac{x^{2}}{4}$ .

$y^{2} = 25 + \frac{- 25 x^{2}}{4}$

Step 3.2.1.4

Move the negative in front of the fraction.

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

Step 4

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

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

Step 5

Simplify $\pm \sqrt{25 - \frac{25 x^{2}}{4}}$ .

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Step 5.1

Write the expression using exponents.

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Step 5.1.1

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

$y = \pm \sqrt{5^{2} - \frac{25 x^{2}}{4}}$

Step 5.1.2

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

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

Step 5.2

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 $b = \frac{5 x}{2}$ .

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

Step 6

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

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Step 6.1

First, use the positive value of the $\pm$ to find the first solution.

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

Step 6.2

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

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

Step 6.3

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

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

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

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

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

Step 7

To write $5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

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

Step 8

Combine $5$ and $\frac{2}{2}$ .

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

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

Step 9

Combine the numerators over the common denominator.

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

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

Step 10

Factor $5$ out of $5 \cdot 2 + 5 x$ .

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Step 10.1

Factor $5$ out of $5 \cdot 2$ .

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

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

Step 10.2

Factor $5$ out of $5 x$ .

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

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

Step 10.3

Factor $5$ out of $5 (2) + 5 (x)$ .

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

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

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

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

Step 11

To write $5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

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

Step 12

Combine $5$ and $\frac{2}{2}$ .

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

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

Step 13

Combine the numerators over the common denominator.

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

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

Step 14

Factor $5$ out of $5 \cdot 2 - 5 x$ .

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Step 14.1

Factor $5$ out of $5 \cdot 2$ .

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

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

Step 14.2

Factor $5$ out of $- 5 x$ .

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

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

Step 14.3

Factor $5$ out of $5 (2) + 5 (- x)$ .

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

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

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

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

Step 15

Multiply $\frac{5 (2 + x)}{2}$ by $\frac{5 (2 - x)}{2}$ .

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

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

Step 16

Multiply $5$ by $5$ .

$y = \sqrt{\frac{25 (2 + x) (2 - x)}{2 \cdot 2}}$

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

Step 17

Multiply $2$ by $2$ .

$y = \sqrt{\frac{25 (2 + x) (2 - x)}{4}}$

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

Step 18

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

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Step 18.1

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

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

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

Step 18.2

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

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

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

Step 18.3

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

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

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

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

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

Step 19

Pull terms out from under the radical.

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

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

Step 20

Combine $\frac{5}{2}$ and $\sqrt{(2 + x) (2 - x)}$ .

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

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

Step 21

To write $5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

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

Step 22

Combine $5$ and $\frac{2}{2}$ .

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

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

Step 23

Combine the numerators over the common denominator.

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

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

Step 24

Factor $5$ out of $5 \cdot 2 + 5 x$ .

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Step 24.1

Factor $5$ out of $5 \cdot 2$ .

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

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

Step 24.2

Factor $5$ out of $5 x$ .

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

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

Step 24.3

Factor $5$ out of $5 (2) + 5 (x)$ .

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

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

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

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

Step 25

To write $5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

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

Step 26

Combine $5$ and $\frac{2}{2}$ .

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

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

Step 27

Combine the numerators over the common denominator.

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

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

Step 28

Factor $5$ out of $5 \cdot 2 - 5 x$ .

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Step 28.1

Factor $5$ out of $5 \cdot 2$ .

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

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

Step 28.2

Factor $5$ out of $- 5 x$ .

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

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

Step 28.3

Factor $5$ out of $5 (2) + 5 (- x)$ .

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

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

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

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

Step 29

Multiply $\frac{5 (2 + x)}{2}$ by $\frac{5 (2 - x)}{2}$ .

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

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

Step 30

Multiply $5$ by $5$ .

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

$y = - \sqrt{\frac{25 (2 + x) (2 - x)}{2 \cdot 2}}$

Step 31

Multiply $2$ by $2$ .

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

$y = - \sqrt{\frac{25 (2 + x) (2 - x)}{4}}$

Step 32

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

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Step 32.1

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

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

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

Step 32.2

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

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

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

Step 32.3

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

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

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

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

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

Step 33

Pull terms out from under the radical.

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

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

Step 34

Combine $\frac{5}{2}$ and $\sqrt{(2 + x) (2 - x)}$ .

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

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

Step 35

The given equation $y = \frac{5 \sqrt{(2 + x) (2 - x)}}{2}, - \frac{5 \sqrt{(2 + x) (2 - x)}}{2}$ can not be written as $y = k x^{2}$ , so $y$ doesn't vary directly with $x^{2}$ .

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