Pre-Algebra Examples

$\frac{x^{2}}{50^{2}} + \frac{y^{2}}{20^{2}} = 1$

Step 1

Simplify each term.

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Step 1.1

Raise $50$ to the power of $2$ .

$\frac{x^{2}}{2500} + \frac{y^{2}}{20^{2}} = 1$

Step 1.2

Raise $20$ to the power of $2$ .

$\frac{x^{2}}{2500} + \frac{y^{2}}{400} = 1$

Step 2

Subtract $\frac{x^{2}}{2500}$ from both sides of the equation.

$\frac{y^{2}}{400} = 1 - \frac{x^{2}}{2500}$

Step 3

Multiply both sides of the equation by $400$ .

$400 \frac{y^{2}}{400} = 400 (1 - \frac{x^{2}}{2500})$

Step 4

Simplify both sides of the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Cancel the common factor of $400$ .

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Step 4.1.1.1

Cancel the common factor.

$400 \frac{y^{2}}{400} = 400 (1 - \frac{x^{2}}{2500})$

Step 4.1.1.2

Rewrite the expression.

$y^{2} = 400 (1 - \frac{x^{2}}{2500})$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $400 (1 - \frac{x^{2}}{2500})$ .

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Step 4.2.1.1

Apply the distributive property.

$y^{2} = 400 \cdot 1 + 400 (- \frac{x^{2}}{2500})$

Step 4.2.1.2

Multiply $400$ by $1$ .

$y^{2} = 400 + 400 (- \frac{x^{2}}{2500})$

Step 4.2.1.3

Cancel the common factor of $100$ .

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Step 4.2.1.3.1

Move the leading negative in $- \frac{x^{2}}{2500}$ into the numerator.

$y^{2} = 400 + 400 \frac{- x^{2}}{2500}$

Step 4.2.1.3.2

Factor $100$ out of $400$ .

$y^{2} = 400 + 100 (4) \frac{- x^{2}}{2500}$

Step 4.2.1.3.3

Factor $100$ out of $2500$ .

$y^{2} = 400 + 100 \cdot 4 \frac{- x^{2}}{100 \cdot 25}$

Step 4.2.1.3.4

Cancel the common factor.

$y^{2} = 400 + 100 \cdot 4 \frac{- x^{2}}{100 \cdot 25}$

Step 4.2.1.3.5

Rewrite the expression.

$y^{2} = 400 + 4 \frac{- x^{2}}{25}$

Step 4.2.1.4

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

$y^{2} = 400 + \frac{4 (- x^{2})}{25}$

Step 4.2.1.5

Simplify the expression.

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Step 4.2.1.5.1

Multiply $- 1$ by $4$ .

$y^{2} = 400 + \frac{- 4 x^{2}}{25}$

Step 4.2.1.5.2

Move the negative in front of the fraction.

$y^{2} = 400 - \frac{4 x^{2}}{25}$

Step 5

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

$y = \pm \sqrt{400 - \frac{4 x^{2}}{25}}$

Step 6

Simplify $\pm \sqrt{400 - \frac{4 x^{2}}{25}}$ .

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Step 6.1

Write the expression using exponents.

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Step 6.1.1

Rewrite $400$ as $20^{2}$ .

$y = \pm \sqrt{20^{2} - \frac{4 x^{2}}{25}}$

Step 6.1.2

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

$y = \pm \sqrt{20^{2} - {(\frac{2 x}{5})}^{2}}$

Step 6.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 20$ and $b = \frac{2 x}{5}$ .

$y = \pm \sqrt{(20 + \frac{2 x}{5}) (20 - \frac{2 x}{5})}$

Step 7

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

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Step 7.1

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

$y = \sqrt{(20 + \frac{2 x}{5}) (20 - \frac{2 x}{5})}$

Step 7.2

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

$y = - \sqrt{(20 + \frac{2 x}{5}) (20 - \frac{2 x}{5})}$

Step 7.3

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

$y = \sqrt{(20 + \frac{2 x}{5}) (20 - \frac{2 x}{5})}$

$y = - \sqrt{(20 + \frac{2 x}{5}) (20 - \frac{2 x}{5})}$

$y = \sqrt{(20 + \frac{2 x}{5}) (20 - \frac{2 x}{5})}$

$y = - \sqrt{(20 + \frac{2 x}{5}) (20 - \frac{2 x}{5})}$

Step 8

To write $20$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y = \sqrt{(20 \cdot \frac{5}{5} + \frac{2 x}{5}) (20 - \frac{2 x}{5})}$

$y = - \sqrt{(20 + \frac{2 x}{5}) (20 - \frac{2 x}{5})}$

Step 9

Combine $20$ and $\frac{5}{5}$ .

$y = \sqrt{(\frac{20 \cdot 5}{5} + \frac{2 x}{5}) (20 - \frac{2 x}{5})}$

$y = - \sqrt{(20 + \frac{2 x}{5}) (20 - \frac{2 x}{5})}$

Step 10

Combine the numerators over the common denominator.

$y = \sqrt{\frac{20 \cdot 5 + 2 x}{5} \cdot (20 - \frac{2 x}{5})}$

$y = - \sqrt{(20 + \frac{2 x}{5}) (20 - \frac{2 x}{5})}$

Step 11

Simplify the numerator.

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Step 11.1

Factor $2$ out of $20 \cdot 5 + 2 x$ .

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Step 11.1.1

Factor $2$ out of $20 \cdot 5$ .

$y = \sqrt{\frac{2 (10 \cdot 5) + 2 x}{5} \cdot (20 - \frac{2 x}{5})}$

$y = - \sqrt{(20 + \frac{2 x}{5}) (20 - \frac{2 x}{5})}$

Step 11.1.2

Factor $2$ out of $2 x$ .

$y = \sqrt{\frac{2 (10 \cdot 5) + 2 (x)}{5} \cdot (20 - \frac{2 x}{5})}$

$y = - \sqrt{(20 + \frac{2 x}{5}) (20 - \frac{2 x}{5})}$

Step 11.1.3

Factor $2$ out of $2 (10 \cdot 5) + 2 (x)$ .

$y = \sqrt{\frac{2 (10 \cdot 5 + x)}{5} \cdot (20 - \frac{2 x}{5})}$

$y = - \sqrt{(20 + \frac{2 x}{5}) (20 - \frac{2 x}{5})}$

$y = \sqrt{\frac{2 (10 \cdot 5 + x)}{5} \cdot (20 - \frac{2 x}{5})}$

$y = - \sqrt{(20 + \frac{2 x}{5}) (20 - \frac{2 x}{5})}$

Step 11.2

Multiply $10$ by $5$ .

$y = \sqrt{\frac{2 (50 + x)}{5} \cdot (20 - \frac{2 x}{5})}$

$y = - \sqrt{(20 + \frac{2 x}{5}) (20 - \frac{2 x}{5})}$

$y = \sqrt{\frac{2 (50 + x)}{5} \cdot (20 - \frac{2 x}{5})}$

$y = - \sqrt{(20 + \frac{2 x}{5}) (20 - \frac{2 x}{5})}$

Step 12

To write $20$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y = \sqrt{\frac{2 (50 + x)}{5} \cdot (20 \cdot \frac{5}{5} - \frac{2 x}{5})}$

$y = - \sqrt{(20 + \frac{2 x}{5}) (20 - \frac{2 x}{5})}$

Step 13

Combine $20$ and $\frac{5}{5}$ .

$y = \sqrt{\frac{2 (50 + x)}{5} \cdot (\frac{20 \cdot 5}{5} - \frac{2 x}{5})}$

$y = - \sqrt{(20 + \frac{2 x}{5}) (20 - \frac{2 x}{5})}$

Step 14

Combine the numerators over the common denominator.

$y = \sqrt{\frac{2 (50 + x)}{5} \cdot \frac{20 \cdot 5 - 2 x}{5}}$

$y = - \sqrt{(20 + \frac{2 x}{5}) (20 - \frac{2 x}{5})}$

Step 15

Simplify the numerator.

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Step 15.1

Factor $2$ out of $20 \cdot 5 - 2 x$ .

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Step 15.1.1

Factor $2$ out of $20 \cdot 5$ .

$y = \sqrt{\frac{2 (50 + x)}{5} \cdot \frac{2 (10 \cdot 5) - 2 x}{5}}$

$y = - \sqrt{(20 + \frac{2 x}{5}) (20 - \frac{2 x}{5})}$

Step 15.1.2

Factor $2$ out of $- 2 x$ .

$y = \sqrt{\frac{2 (50 + x)}{5} \cdot \frac{2 (10 \cdot 5) + 2 (- x)}{5}}$

$y = - \sqrt{(20 + \frac{2 x}{5}) (20 - \frac{2 x}{5})}$

Step 15.1.3

Factor $2$ out of $2 (10 \cdot 5) + 2 (- x)$ .

$y = \sqrt{\frac{2 (50 + x)}{5} \cdot \frac{2 (10 \cdot 5 - x)}{5}}$

$y = - \sqrt{(20 + \frac{2 x}{5}) (20 - \frac{2 x}{5})}$

$y = \sqrt{\frac{2 (50 + x)}{5} \cdot \frac{2 (10 \cdot 5 - x)}{5}}$

$y = - \sqrt{(20 + \frac{2 x}{5}) (20 - \frac{2 x}{5})}$

Step 15.2

Multiply $10$ by $5$ .

$y = \sqrt{\frac{2 (50 + x)}{5} \cdot \frac{2 (50 - x)}{5}}$

$y = - \sqrt{(20 + \frac{2 x}{5}) (20 - \frac{2 x}{5})}$

$y = \sqrt{\frac{2 (50 + x)}{5} \cdot \frac{2 (50 - x)}{5}}$

$y = - \sqrt{(20 + \frac{2 x}{5}) (20 - \frac{2 x}{5})}$

Step 16

Multiply $\frac{2 (50 + x)}{5}$ by $\frac{2 (50 - x)}{5}$ .

$y = \sqrt{\frac{2 (50 + x) (2 (50 - x))}{5 \cdot 5}}$

$y = - \sqrt{(20 + \frac{2 x}{5}) (20 - \frac{2 x}{5})}$

Step 17

Multiply $2$ by $2$ .

$y = \sqrt{\frac{4 (50 + x) (50 - x)}{5 \cdot 5}}$

$y = - \sqrt{(20 + \frac{2 x}{5}) (20 - \frac{2 x}{5})}$

Step 18

Multiply $5$ by $5$ .

$y = \sqrt{\frac{4 (50 + x) (50 - x)}{25}}$

$y = - \sqrt{(20 + \frac{2 x}{5}) (20 - \frac{2 x}{5})}$

Step 19

Rewrite $\frac{4 (50 + x) (50 - x)}{25}$ as ${(\frac{2}{5})}^{2} ((50 + x) (50 - x))$ .

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Step 19.1

Factor the perfect power $2^{2}$ out of $4 (50 + x) (50 - x)$ .

$y = \sqrt{\frac{2^{2} ((50 + x) (50 - x))}{25}}$

$y = - \sqrt{(20 + \frac{2 x}{5}) (20 - \frac{2 x}{5})}$

Step 19.2

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

$y = \sqrt{\frac{2^{2} ((50 + x) (50 - x))}{5^{2} \cdot 1}}$

$y = - \sqrt{(20 + \frac{2 x}{5}) (20 - \frac{2 x}{5})}$

Step 19.3

Rearrange the fraction $\frac{2^{2} ((50 + x) (50 - x))}{5^{2} \cdot 1}$ .

$y = \sqrt{{(\frac{2}{5})}^{2} ((50 + x) (50 - x))}$

$y = - \sqrt{(20 + \frac{2 x}{5}) (20 - \frac{2 x}{5})}$

$y = \sqrt{{(\frac{2}{5})}^{2} ((50 + x) (50 - x))}$

$y = - \sqrt{(20 + \frac{2 x}{5}) (20 - \frac{2 x}{5})}$

Step 20

Pull terms out from under the radical.

$y = \frac{2}{5} \cdot \sqrt{(50 + x) (50 - x)}$

$y = - \sqrt{(20 + \frac{2 x}{5}) (20 - \frac{2 x}{5})}$

Step 21

Combine $\frac{2}{5}$ and $\sqrt{(50 + x) (50 - x)}$ .

$y = \frac{2 \sqrt{(50 + x) (50 - x)}}{5}$

$y = - \sqrt{(20 + \frac{2 x}{5}) (20 - \frac{2 x}{5})}$

Step 22

To write $20$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y = \frac{2 \sqrt{(50 + x) (50 - x)}}{5}$

$y = - \sqrt{(20 \cdot \frac{5}{5} + \frac{2 x}{5}) (20 - \frac{2 x}{5})}$

Step 23

Combine $20$ and $\frac{5}{5}$ .

$y = \frac{2 \sqrt{(50 + x) (50 - x)}}{5}$

$y = - \sqrt{(\frac{20 \cdot 5}{5} + \frac{2 x}{5}) (20 - \frac{2 x}{5})}$

Step 24

Combine the numerators over the common denominator.

$y = \frac{2 \sqrt{(50 + x) (50 - x)}}{5}$

$y = - \sqrt{\frac{20 \cdot 5 + 2 x}{5} \cdot (20 - \frac{2 x}{5})}$

Step 25

Simplify the numerator.

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Step 25.1

Factor $2$ out of $20 \cdot 5 + 2 x$ .

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Step 25.1.1

Factor $2$ out of $20 \cdot 5$ .

$y = \frac{2 \sqrt{(50 + x) (50 - x)}}{5}$

$y = - \sqrt{\frac{2 (10 \cdot 5) + 2 x}{5} \cdot (20 - \frac{2 x}{5})}$

Step 25.1.2

Factor $2$ out of $2 x$ .

$y = \frac{2 \sqrt{(50 + x) (50 - x)}}{5}$

$y = - \sqrt{\frac{2 (10 \cdot 5) + 2 (x)}{5} \cdot (20 - \frac{2 x}{5})}$

Step 25.1.3

Factor $2$ out of $2 (10 \cdot 5) + 2 (x)$ .

$y = \frac{2 \sqrt{(50 + x) (50 - x)}}{5}$

$y = - \sqrt{\frac{2 (10 \cdot 5 + x)}{5} \cdot (20 - \frac{2 x}{5})}$

$y = \frac{2 \sqrt{(50 + x) (50 - x)}}{5}$

$y = - \sqrt{\frac{2 (10 \cdot 5 + x)}{5} \cdot (20 - \frac{2 x}{5})}$

Step 25.2

Multiply $10$ by $5$ .

$y = \frac{2 \sqrt{(50 + x) (50 - x)}}{5}$

$y = - \sqrt{\frac{2 (50 + x)}{5} \cdot (20 - \frac{2 x}{5})}$

$y = \frac{2 \sqrt{(50 + x) (50 - x)}}{5}$

$y = - \sqrt{\frac{2 (50 + x)}{5} \cdot (20 - \frac{2 x}{5})}$

Step 26

To write $20$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y = \frac{2 \sqrt{(50 + x) (50 - x)}}{5}$

$y = - \sqrt{\frac{2 (50 + x)}{5} \cdot (20 \cdot \frac{5}{5} - \frac{2 x}{5})}$

Step 27

Combine $20$ and $\frac{5}{5}$ .

$y = \frac{2 \sqrt{(50 + x) (50 - x)}}{5}$

$y = - \sqrt{\frac{2 (50 + x)}{5} \cdot (\frac{20 \cdot 5}{5} - \frac{2 x}{5})}$

Step 28

Combine the numerators over the common denominator.

$y = \frac{2 \sqrt{(50 + x) (50 - x)}}{5}$

$y = - \sqrt{\frac{2 (50 + x)}{5} \cdot \frac{20 \cdot 5 - 2 x}{5}}$

Step 29

Simplify the numerator.

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Step 29.1

Factor $2$ out of $20 \cdot 5 - 2 x$ .

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Step 29.1.1

Factor $2$ out of $20 \cdot 5$ .

$y = \frac{2 \sqrt{(50 + x) (50 - x)}}{5}$

$y = - \sqrt{\frac{2 (50 + x)}{5} \cdot \frac{2 (10 \cdot 5) - 2 x}{5}}$

Step 29.1.2

Factor $2$ out of $- 2 x$ .

$y = \frac{2 \sqrt{(50 + x) (50 - x)}}{5}$

$y = - \sqrt{\frac{2 (50 + x)}{5} \cdot \frac{2 (10 \cdot 5) + 2 (- x)}{5}}$

Step 29.1.3

Factor $2$ out of $2 (10 \cdot 5) + 2 (- x)$ .

$y = \frac{2 \sqrt{(50 + x) (50 - x)}}{5}$

$y = - \sqrt{\frac{2 (50 + x)}{5} \cdot \frac{2 (10 \cdot 5 - x)}{5}}$

$y = \frac{2 \sqrt{(50 + x) (50 - x)}}{5}$

$y = - \sqrt{\frac{2 (50 + x)}{5} \cdot \frac{2 (10 \cdot 5 - x)}{5}}$

Step 29.2

Multiply $10$ by $5$ .

$y = \frac{2 \sqrt{(50 + x) (50 - x)}}{5}$

$y = - \sqrt{\frac{2 (50 + x)}{5} \cdot \frac{2 (50 - x)}{5}}$

$y = \frac{2 \sqrt{(50 + x) (50 - x)}}{5}$

$y = - \sqrt{\frac{2 (50 + x)}{5} \cdot \frac{2 (50 - x)}{5}}$

Step 30

Multiply $\frac{2 (50 + x)}{5}$ by $\frac{2 (50 - x)}{5}$ .

$y = \frac{2 \sqrt{(50 + x) (50 - x)}}{5}$

$y = - \sqrt{\frac{2 (50 + x) (2 (50 - x))}{5 \cdot 5}}$

Step 31

Multiply $2$ by $2$ .

$y = \frac{2 \sqrt{(50 + x) (50 - x)}}{5}$

$y = - \sqrt{\frac{4 (50 + x) (50 - x)}{5 \cdot 5}}$

Step 32

Multiply $5$ by $5$ .

$y = \frac{2 \sqrt{(50 + x) (50 - x)}}{5}$

$y = - \sqrt{\frac{4 (50 + x) (50 - x)}{25}}$

Step 33

Rewrite $\frac{4 (50 + x) (50 - x)}{25}$ as ${(\frac{2}{5})}^{2} ((50 + x) (50 - x))$ .

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Step 33.1

Factor the perfect power $2^{2}$ out of $4 (50 + x) (50 - x)$ .

$y = \frac{2 \sqrt{(50 + x) (50 - x)}}{5}$

$y = - \sqrt{\frac{2^{2} ((50 + x) (50 - x))}{25}}$

Step 33.2

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

$y = \frac{2 \sqrt{(50 + x) (50 - x)}}{5}$

$y = - \sqrt{\frac{2^{2} ((50 + x) (50 - x))}{5^{2} \cdot 1}}$

Step 33.3

Rearrange the fraction $\frac{2^{2} ((50 + x) (50 - x))}{5^{2} \cdot 1}$ .

$y = \frac{2 \sqrt{(50 + x) (50 - x)}}{5}$

$y = - \sqrt{{(\frac{2}{5})}^{2} ((50 + x) (50 - x))}$

$y = \frac{2 \sqrt{(50 + x) (50 - x)}}{5}$

$y = - \sqrt{{(\frac{2}{5})}^{2} ((50 + x) (50 - x))}$

Step 34

Pull terms out from under the radical.

$y = \frac{2 \sqrt{(50 + x) (50 - x)}}{5}$

$y = - (\frac{2}{5} \cdot \sqrt{(50 + x) (50 - x)})$

Step 35

Combine $\frac{2}{5}$ and $\sqrt{(50 + x) (50 - x)}$ .

$y = \frac{2 \sqrt{(50 + x) (50 - x)}}{5}$

$y = - \frac{2 \sqrt{(50 + x) (50 - x)}}{5}$

Step 36

The given equation $y = \frac{2 \sqrt{(50 + x) (50 - x)}}{5}, - \frac{2 \sqrt{(50 + x) (50 - x)}}{5}$ can not be written as $y = k x^{2}$ , so $y$ doesn't vary directly with $x^{2}$ .

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